| L(s) = 1 | + 2·9-s − 8·11-s + 4·29-s − 20·37-s + 8·43-s − 12·53-s − 24·67-s + 16·79-s − 5·81-s − 16·99-s + 24·107-s + 4·109-s − 20·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.41·11-s + 0.742·29-s − 3.28·37-s + 1.21·43-s − 1.64·53-s − 2.93·67-s + 1.80·79-s − 5/9·81-s − 1.60·99-s + 2.32·107-s + 0.383·109-s − 1.88·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 162 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42663358761105459517047514726, −7.24699762226719052833511868071, −7.00289245424339179904598690270, −6.45868569226193103237627933297, −6.03783930016312035034650127645, −5.89470841007552009305975690070, −5.33137476941517195459384292241, −4.98811988168643011422121142972, −4.87978485716341105580617482908, −4.56001945034488189843162739891, −3.93015100925216601171261742105, −3.60689868453950293168379237081, −3.13606076284938619817993462843, −2.82946370345440302101898039150, −2.44615600636017000275411183114, −1.93584920541769368439291033202, −1.56908992681153513783088277374, −0.965135481410223608797873410382, 0, 0,
0.965135481410223608797873410382, 1.56908992681153513783088277374, 1.93584920541769368439291033202, 2.44615600636017000275411183114, 2.82946370345440302101898039150, 3.13606076284938619817993462843, 3.60689868453950293168379237081, 3.93015100925216601171261742105, 4.56001945034488189843162739891, 4.87978485716341105580617482908, 4.98811988168643011422121142972, 5.33137476941517195459384292241, 5.89470841007552009305975690070, 6.03783930016312035034650127645, 6.45868569226193103237627933297, 7.00289245424339179904598690270, 7.24699762226719052833511868071, 7.42663358761105459517047514726
