| L(s) = 1 | − 4-s − 9-s + 16-s + 20·29-s + 36-s + 4·41-s + 14·49-s + 12·61-s − 64-s + 32·71-s + 81-s + 12·89-s + 12·101-s + 28·109-s − 20·116-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s + 3.71·29-s + 1/6·36-s + 0.624·41-s + 2·49-s + 1.53·61-s − 1/8·64-s + 3.79·71-s + 1/9·81-s + 1.27·89-s + 1.19·101-s + 2.68·109-s − 1.85·116-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 0.312·164-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489899517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489899517\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.324829414365897403534583934842, −8.873004737428168776880637713402, −8.616922852978376070472245639819, −8.345238335244018553032912949153, −7.76155103767951533411174428588, −7.67974513181660500879309604141, −6.79079837994010600856173324279, −6.68956492523292333560510877869, −6.33601106139981271770406232860, −5.68444626772792164123977746234, −5.38195764772268698999721999460, −4.89853320263521452178116321586, −4.47907657810209676321139527920, −4.15707191570699309875231884706, −3.45813428929516679023275408846, −3.16236242952289672021194959278, −2.34576781878962605888678853675, −2.24604823741025748567118278636, −0.872231426305974713810506079661, −0.837038914943169770057565526587,
0.837038914943169770057565526587, 0.872231426305974713810506079661, 2.24604823741025748567118278636, 2.34576781878962605888678853675, 3.16236242952289672021194959278, 3.45813428929516679023275408846, 4.15707191570699309875231884706, 4.47907657810209676321139527920, 4.89853320263521452178116321586, 5.38195764772268698999721999460, 5.68444626772792164123977746234, 6.33601106139981271770406232860, 6.68956492523292333560510877869, 6.79079837994010600856173324279, 7.67974513181660500879309604141, 7.76155103767951533411174428588, 8.345238335244018553032912949153, 8.616922852978376070472245639819, 8.873004737428168776880637713402, 9.324829414365897403534583934842
