| L(s) = 1 | + 5-s + 3·7-s − 3·11-s + 8·17-s − 12·19-s + 6·23-s + 5·25-s + 2·29-s + 9·31-s + 3·35-s − 4·37-s + 10·41-s + 6·43-s + 6·47-s + 7·49-s + 26·53-s − 3·55-s − 12·59-s − 8·61-s + 6·67-s − 24·71-s + 18·73-s − 9·77-s − 3·83-s + 8·85-s + 28·89-s − 12·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.904·11-s + 1.94·17-s − 2.75·19-s + 1.25·23-s + 25-s + 0.371·29-s + 1.61·31-s + 0.507·35-s − 0.657·37-s + 1.56·41-s + 0.914·43-s + 0.875·47-s + 49-s + 3.57·53-s − 0.404·55-s − 1.56·59-s − 1.02·61-s + 0.733·67-s − 2.84·71-s + 2.10·73-s − 1.02·77-s − 0.329·83-s + 0.867·85-s + 2.96·89-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.962750267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.962750267\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.841060950394308479581827706312, −8.819346814604100255739101658364, −8.226120024367432596341438664645, −8.075839801156415310934025897735, −7.48196711426851802790243109007, −7.34236234737830867441348407904, −6.81258704366845437877083106946, −6.30550073494267148257715964716, −5.83947962666259306511394282681, −5.72556258867310486551437243281, −5.02501958303890848718776280880, −4.87361384513441165614587573010, −4.24956306862705455162588548021, −4.15336891468425782759188445146, −3.20138194646257024700427056504, −2.90752481199080668703383881696, −2.16219155677443048109132050131, −2.15595654771543905306353677548, −1.02292980729207500018704471317, −0.820894858939716181770140725296,
0.820894858939716181770140725296, 1.02292980729207500018704471317, 2.15595654771543905306353677548, 2.16219155677443048109132050131, 2.90752481199080668703383881696, 3.20138194646257024700427056504, 4.15336891468425782759188445146, 4.24956306862705455162588548021, 4.87361384513441165614587573010, 5.02501958303890848718776280880, 5.72556258867310486551437243281, 5.83947962666259306511394282681, 6.30550073494267148257715964716, 6.81258704366845437877083106946, 7.34236234737830867441348407904, 7.48196711426851802790243109007, 8.075839801156415310934025897735, 8.226120024367432596341438664645, 8.819346814604100255739101658364, 8.841060950394308479581827706312
