| L(s) = 1 | + 4·4-s + 4·16-s − 68·25-s + 128·37-s + 160·43-s − 16·64-s + 192·67-s + 592·79-s − 272·100-s + 344·109-s − 468·121-s + 127-s + 131-s + 137-s + 139-s + 512·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 704·169-s + 640·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s + 1/4·16-s − 2.71·25-s + 3.45·37-s + 3.72·43-s − 1/4·64-s + 2.86·67-s + 7.49·79-s − 2.71·100-s + 3.15·109-s − 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.45·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.16·169-s + 3.72·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(18.32637386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.32637386\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 34 T^{2} + 531 T^{4} + 34 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 234 T^{2} + 40115 T^{4} + 234 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 176 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 562 T^{2} + 232323 T^{4} + 562 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 210 T^{2} - 86221 T^{4} - 210 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 294 T^{2} - 193405 T^{4} - 294 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 624 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 670 T^{2} - 474621 T^{4} + 670 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 32 T - 345 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 1918 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 20 T + p^{2} T^{2} )^{8} \) |
| 47 | \( ( 1 + 4018 T^{2} + 11264643 T^{4} + 4018 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3360 T^{2} + 3399119 T^{4} - 3360 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6946 T^{2} + 36129555 T^{4} + 6946 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 480 T^{2} - 13615441 T^{4} - 480 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 48 T - 2185 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 4250 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 3792 T^{2} - 14018977 T^{4} + 3792 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 148 T + 15663 T^{2} - 148 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 7378 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 4606 T^{2} - 41527005 T^{4} + 4606 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4944 T^{2} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.09923496087980252722804606771, −3.98322815221639811342235467661, −3.79721280219042121146355296154, −3.59358225982962693548724660193, −3.59277670969458377539430381065, −3.54934514290746481322033780878, −3.52668784338878938449012555168, −3.27282942724270726807117393762, −2.84720030382197241611708606762, −2.79395526292825170200389709741, −2.70639543852417181851246440122, −2.46436587660661309979746228995, −2.38984525078270648995136636718, −2.19801082735524636854691465706, −2.18538957674873460122643813065, −2.14294275313713335684140583584, −2.02468000817191919363043446369, −1.47286277187066860060703507308, −1.38189409192808109422983825483, −1.34667483154596513131380278529, −0.931166697341004721690758423992, −0.848530430226665491279746872635, −0.58062584445588649405884402276, −0.50733187373407850712237596916, −0.28184435681888848515937201656,
0.28184435681888848515937201656, 0.50733187373407850712237596916, 0.58062584445588649405884402276, 0.848530430226665491279746872635, 0.931166697341004721690758423992, 1.34667483154596513131380278529, 1.38189409192808109422983825483, 1.47286277187066860060703507308, 2.02468000817191919363043446369, 2.14294275313713335684140583584, 2.18538957674873460122643813065, 2.19801082735524636854691465706, 2.38984525078270648995136636718, 2.46436587660661309979746228995, 2.70639543852417181851246440122, 2.79395526292825170200389709741, 2.84720030382197241611708606762, 3.27282942724270726807117393762, 3.52668784338878938449012555168, 3.54934514290746481322033780878, 3.59277670969458377539430381065, 3.59358225982962693548724660193, 3.79721280219042121146355296154, 3.98322815221639811342235467661, 4.09923496087980252722804606771
Plot not available for L-functions of degree greater than 10.
