| L(s) = 1 | + 20·5-s + 20·9-s − 28·13-s − 8·17-s + 164·25-s + 104·29-s + 8·37-s − 8·41-s + 400·45-s − 14·49-s − 176·53-s − 60·61-s − 560·65-s + 104·73-s + 166·81-s − 160·85-s − 376·89-s + 152·97-s + 92·101-s + 136·109-s + 232·113-s − 560·117-s + 420·121-s + 420·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4·5-s + 20/9·9-s − 2.15·13-s − 0.470·17-s + 6.55·25-s + 3.58·29-s + 8/37·37-s − 0.195·41-s + 80/9·45-s − 2/7·49-s − 3.32·53-s − 0.983·61-s − 8.61·65-s + 1.42·73-s + 2.04·81-s − 1.88·85-s − 4.22·89-s + 1.56·97-s + 0.910·101-s + 1.24·109-s + 2.05·113-s − 4.78·117-s + 3.47·121-s + 3.35·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.460411681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.460411681\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 26 p^{2} T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 2 p T + 68 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 420 T^{2} + 72934 T^{4} - 420 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 14 T + 380 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 470 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1092 T^{2} + 554026 T^{4} - 1092 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1212 T^{2} + 919750 T^{4} - 1212 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 52 T + 1658 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2916 T^{2} + 3944134 T^{4} - 2916 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 2042 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 566 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2460 T^{2} + 7801702 T^{4} + 2460 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3204 T^{2} + 4985734 T^{4} - 3204 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 88 T + 4754 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3268 T^{2} + 16106730 T^{4} - 3268 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 30 T + 2564 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4572 T^{2} + 42660838 T^{4} + 4572 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 15556 T^{2} + 106095174 T^{4} - 15556 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 52 T + 10326 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 19332 T^{2} + 170120326 T^{4} - 19332 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12084 T^{2} + 87882154 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 188 T + 22886 T^{2} + 188 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 76 T + 4134 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.44990094228616044432552793610, −6.12643116952793548866500126192, −6.09326242825342945035526463634, −5.94633398058156112445404046055, −5.55763689527959211576316316573, −5.30972534680707878769157368856, −5.12698478439188602988798733221, −4.83992709849632712735622934590, −4.79012196136992620848652491919, −4.56427454116207334198979592852, −4.39584645859102398371939933949, −4.26445612382055476728524434658, −3.70459052678609179832332274468, −3.42583003087239976774334606890, −2.99654359252649476666792388256, −2.86104576488333320469614798219, −2.72200312017742615889631375941, −2.19939077874711277068547321464, −2.10130620052246721662964381066, −2.01878160350550283453259973533, −1.70423143059432881757157120165, −1.43610805739609478714910940956, −1.12401639208688344899247311470, −0.906779661544205815796493179844, −0.23549408420363284333212681368,
0.23549408420363284333212681368, 0.906779661544205815796493179844, 1.12401639208688344899247311470, 1.43610805739609478714910940956, 1.70423143059432881757157120165, 2.01878160350550283453259973533, 2.10130620052246721662964381066, 2.19939077874711277068547321464, 2.72200312017742615889631375941, 2.86104576488333320469614798219, 2.99654359252649476666792388256, 3.42583003087239976774334606890, 3.70459052678609179832332274468, 4.26445612382055476728524434658, 4.39584645859102398371939933949, 4.56427454116207334198979592852, 4.79012196136992620848652491919, 4.83992709849632712735622934590, 5.12698478439188602988798733221, 5.30972534680707878769157368856, 5.55763689527959211576316316573, 5.94633398058156112445404046055, 6.09326242825342945035526463634, 6.12643116952793548866500126192, 6.44990094228616044432552793610
