| L(s) = 1 | + 4·4-s − 26·13-s + 3·16-s − 26·19-s + 22·25-s − 22·31-s + 146·37-s + 54·43-s − 104·52-s + 136·61-s + 46·64-s − 2·67-s + 482·73-s − 104·76-s − 42·79-s + 284·97-s + 88·100-s − 234·103-s − 130·109-s + 514·121-s − 88·124-s + 127-s + 131-s + 137-s + 139-s + 584·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 2·13-s + 3/16·16-s − 1.36·19-s + 0.879·25-s − 0.709·31-s + 3.94·37-s + 1.25·43-s − 2·52-s + 2.22·61-s + 0.718·64-s − 0.0298·67-s + 6.60·73-s − 1.36·76-s − 0.531·79-s + 2.92·97-s + 0.879·100-s − 2.27·103-s − 1.19·109-s + 4.24·121-s − 0.709·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.94·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.275479756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.275479756\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + 13 T^{4} - 43 p T^{6} + 13 p^{4} T^{8} - p^{10} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 - 22 T^{2} + 1132 T^{4} - 17132 T^{6} + 1132 p^{4} T^{8} - 22 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( 1 - 514 T^{2} + 129484 T^{4} - 19636340 T^{6} + 129484 p^{4} T^{8} - 514 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 + p T + 269 T^{2} + 1552 T^{3} + 269 p^{2} T^{4} + p^{5} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 738 T^{2} + 399555 T^{4} - 131047364 T^{6} + 399555 p^{4} T^{8} - 738 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 + 13 T + 845 T^{2} + 6544 T^{3} + 845 p^{2} T^{4} + 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 1890 T^{2} + 1918755 T^{4} - 1249145156 T^{6} + 1918755 p^{4} T^{8} - 1890 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 2920 T^{2} + 4462552 T^{4} - 4500431378 T^{6} + 4462552 p^{4} T^{8} - 2920 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 + 11 T + 1672 T^{2} + 31873 T^{3} + 1672 p^{2} T^{4} + 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - 73 T + 5101 T^{2} - 186980 T^{3} + 5101 p^{2} T^{4} - 73 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 - 6892 T^{2} + 23337223 T^{4} - 49003439192 T^{6} + 23337223 p^{4} T^{8} - 6892 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 - 27 T + 4581 T^{2} - 73568 T^{3} + 4581 p^{2} T^{4} - 27 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 5616 T^{2} + 20345283 T^{4} - 53758551008 T^{6} + 20345283 p^{4} T^{8} - 5616 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 10584 T^{2} + 54376488 T^{4} - 182665815122 T^{6} + 54376488 p^{4} T^{8} - 10584 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 12708 T^{2} + 73521072 T^{4} - 288183912842 T^{6} + 73521072 p^{4} T^{8} - 12708 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 68 T + 8993 T^{2} - 389240 T^{3} + 8993 p^{2} T^{4} - 68 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 + T + 12563 T^{2} + 4546 T^{3} + 12563 p^{2} T^{4} + p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 3984 T^{2} + 20461731 T^{4} - 164780765984 T^{6} + 20461731 p^{4} T^{8} - 3984 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 241 T + 26347 T^{2} - 2035850 T^{3} + 26347 p^{2} T^{4} - 241 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 + 21 T + 3912 T^{2} + 849823 T^{3} + 3912 p^{2} T^{4} + 21 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( 1 - 21858 T^{2} + 183628284 T^{4} - 1109191776932 T^{6} + 183628284 p^{4} T^{8} - 21858 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 - 36766 T^{2} + 634804639 T^{4} - 6402279105860 T^{6} + 634804639 p^{4} T^{8} - 36766 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 - 142 T + 22732 T^{2} - 2546324 T^{3} + 22732 p^{2} T^{4} - 142 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.66720780029857102857397242703, −5.65381015428633125810357108948, −5.61413797424717943238256146484, −5.18384383130832457181311185533, −4.92498296755133034368024134756, −4.90151798147303601544273942777, −4.80601803818945043886471276618, −4.43778994008406293266600489227, −4.31258806719773222697704524787, −4.18385078232472359819493373119, −3.86741688943618149507894721762, −3.74741786142375008310072474458, −3.52302004716550299475403094458, −3.21781874847381792199379213120, −2.86200451278831335336452754547, −2.85656245291352192112796177888, −2.47373107239504648031477874370, −2.30082580366801167973009661514, −2.18318390834975015297843550253, −2.03305648009274683716474670959, −1.89210183602700134503102799170, −1.12327379834690191776037340837, −0.845225446231367220264038746082, −0.74217461916102508802856629128, −0.34626053640919818924491202546,
0.34626053640919818924491202546, 0.74217461916102508802856629128, 0.845225446231367220264038746082, 1.12327379834690191776037340837, 1.89210183602700134503102799170, 2.03305648009274683716474670959, 2.18318390834975015297843550253, 2.30082580366801167973009661514, 2.47373107239504648031477874370, 2.85656245291352192112796177888, 2.86200451278831335336452754547, 3.21781874847381792199379213120, 3.52302004716550299475403094458, 3.74741786142375008310072474458, 3.86741688943618149507894721762, 4.18385078232472359819493373119, 4.31258806719773222697704524787, 4.43778994008406293266600489227, 4.80601803818945043886471276618, 4.90151798147303601544273942777, 4.92498296755133034368024134756, 5.18384383130832457181311185533, 5.61413797424717943238256146484, 5.65381015428633125810357108948, 5.66720780029857102857397242703
Plot not available for L-functions of degree greater than 10.
