| L(s) = 1 | + 4·4-s + 2·5-s + 2·11-s + 12·16-s + 8·19-s + 8·20-s − 25-s − 16·29-s − 8·31-s + 18·41-s + 8·44-s + 13·49-s + 4·55-s + 8·59-s − 22·61-s + 32·64-s + 2·71-s + 32·76-s − 2·79-s + 24·80-s − 30·89-s + 16·95-s − 4·100-s − 20·101-s − 24·109-s − 64·116-s − 19·121-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.894·5-s + 0.603·11-s + 3·16-s + 1.83·19-s + 1.78·20-s − 1/5·25-s − 2.97·29-s − 1.43·31-s + 2.81·41-s + 1.20·44-s + 13/7·49-s + 0.539·55-s + 1.04·59-s − 2.81·61-s + 4·64-s + 0.237·71-s + 3.67·76-s − 0.225·79-s + 2.68·80-s − 3.17·89-s + 1.64·95-s − 2/5·100-s − 1.99·101-s − 2.29·109-s − 5.94·116-s − 1.72·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.057353296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.057353296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−10.87898038833250026621179501557, −10.74608533075263124168735521644, −10.04292145615842887380211661582, −9.609188184575568261816030842822, −9.213494949575643185527614767245, −9.068962347645527030060137420226, −7.85918238551724750336449828649, −7.78005249123483324009133600247, −7.14695892530368312169560811471, −7.14552969028746938793910541669, −6.35180023307200811647785647349, −5.82638549570416249479735915768, −5.51414496080899557448061729831, −5.43827603836401712548631717091, −3.89981107143308988982460920509, −3.87484069815032796321291378078, −2.83980503458216305822408503773, −2.54516969432786931655770227564, −1.67750185880424487464597232158, −1.35212561800511936133670034678,
1.35212561800511936133670034678, 1.67750185880424487464597232158, 2.54516969432786931655770227564, 2.83980503458216305822408503773, 3.87484069815032796321291378078, 3.89981107143308988982460920509, 5.43827603836401712548631717091, 5.51414496080899557448061729831, 5.82638549570416249479735915768, 6.35180023307200811647785647349, 7.14552969028746938793910541669, 7.14695892530368312169560811471, 7.78005249123483324009133600247, 7.85918238551724750336449828649, 9.068962347645527030060137420226, 9.213494949575643185527614767245, 9.609188184575568261816030842822, 10.04292145615842887380211661582, 10.74608533075263124168735521644, 10.87898038833250026621179501557
