| L(s) = 1 | − 4-s + 4·5-s − 9-s + 8·11-s + 16-s − 4·19-s − 4·20-s + 11·25-s + 16·29-s + 36-s − 4·41-s − 8·44-s − 4·45-s + 10·49-s + 32·55-s − 28·59-s − 24·61-s − 64-s + 4·76-s + 16·79-s + 4·80-s + 81-s + 20·89-s − 16·95-s − 8·99-s − 11·100-s − 12·101-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 0.917·19-s − 0.894·20-s + 11/5·25-s + 2.97·29-s + 1/6·36-s − 0.624·41-s − 1.20·44-s − 0.596·45-s + 10/7·49-s + 4.31·55-s − 3.64·59-s − 3.07·61-s − 1/8·64-s + 0.458·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 2.11·89-s − 1.64·95-s − 0.804·99-s − 1.09·100-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.391678409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.391678409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.875538461991572270039562668007, −9.575870782330406636055930869150, −9.192659590786573781069738731393, −8.917048079349350748800251563254, −8.712317486699673711262015630400, −8.137299122905775590988903394213, −7.59514694031345826863883307751, −6.89494957778772227203001286030, −6.39288926976370005862739861098, −6.29783850258393869721708026030, −6.13262135834501838138314047609, −5.35507438991081208800447699704, −4.81442077210364497922619153180, −4.46295378969176626891892718463, −4.05192271090513651256314678160, −3.13246285146005863052316712149, −2.92157440145278415652892382268, −1.97868340288436899675780708410, −1.52298635124479814406354454352, −0.904483457506386688431449941298,
0.904483457506386688431449941298, 1.52298635124479814406354454352, 1.97868340288436899675780708410, 2.92157440145278415652892382268, 3.13246285146005863052316712149, 4.05192271090513651256314678160, 4.46295378969176626891892718463, 4.81442077210364497922619153180, 5.35507438991081208800447699704, 6.13262135834501838138314047609, 6.29783850258393869721708026030, 6.39288926976370005862739861098, 6.89494957778772227203001286030, 7.59514694031345826863883307751, 8.137299122905775590988903394213, 8.712317486699673711262015630400, 8.917048079349350748800251563254, 9.192659590786573781069738731393, 9.575870782330406636055930869150, 9.875538461991572270039562668007
