| L(s) = 1 | + 2·4-s − 9-s + 3·16-s − 2·36-s − 2·49-s − 4·61-s + 4·64-s + 4·79-s + 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + ⋯ |
| L(s) = 1 | + 2·4-s − 9-s + 3·16-s − 2·36-s − 2·49-s − 4·61-s + 4·64-s + 4·79-s + 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.490131892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490131892\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.46020180672746986598322320198, −10.25459188500847120213976126290, −9.600924897993804394861135573876, −9.145070378795872887496922649413, −8.841380063270587955574540273338, −7.937506276045764260070505889456, −7.87464050505085899264970070378, −7.71768273400817361388107516167, −6.87859619786778195954961749199, −6.59237856878386894559391487066, −6.08149711375738883685380744832, −6.07656575759035098726143849334, −5.13072896459162593903227548049, −5.06959454151953004867126607954, −4.07246558130023193839670554749, −3.35201431561297263149031426010, −3.11187708242688814260298900724, −2.53427834112161155281039054535, −1.96214344693111957840067107755, −1.32628486048628880277635760772,
1.32628486048628880277635760772, 1.96214344693111957840067107755, 2.53427834112161155281039054535, 3.11187708242688814260298900724, 3.35201431561297263149031426010, 4.07246558130023193839670554749, 5.06959454151953004867126607954, 5.13072896459162593903227548049, 6.07656575759035098726143849334, 6.08149711375738883685380744832, 6.59237856878386894559391487066, 6.87859619786778195954961749199, 7.71768273400817361388107516167, 7.87464050505085899264970070378, 7.937506276045764260070505889456, 8.841380063270587955574540273338, 9.145070378795872887496922649413, 9.600924897993804394861135573876, 10.25459188500847120213976126290, 10.46020180672746986598322320198
