| L(s) = 1 | − 3-s − 2·4-s + 2·7-s + 9-s + 2·12-s + 4·16-s − 2·21-s − 7·25-s − 27-s − 4·28-s + 4·29-s − 2·36-s + 16·43-s − 4·48-s − 7·49-s − 24·53-s − 4·59-s − 4·61-s + 2·63-s − 8·64-s + 12·71-s + 6·73-s + 7·75-s + 81-s + 4·84-s − 4·87-s − 8·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.577·12-s + 16-s − 0.436·21-s − 7/5·25-s − 0.192·27-s − 0.755·28-s + 0.742·29-s − 1/3·36-s + 2.43·43-s − 0.577·48-s − 49-s − 3.29·53-s − 0.520·59-s − 0.512·61-s + 0.251·63-s − 64-s + 1.42·71-s + 0.702·73-s + 0.808·75-s + 1/9·81-s + 0.436·84-s − 0.428·87-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−8.127471297869059558051546284898, −7.69800230033996657873823491182, −7.57268670837969735844538288727, −6.64441444704426775177764403722, −6.30044424957163342752387110170, −5.79738003881247977630639495666, −5.36701953076705401568809187494, −4.76158685629415364500613814436, −4.57160537934393565707213291122, −3.97468086120425923627726323610, −3.44449310979811290848798183696, −2.67539943625420951049270226971, −1.79189098456849711197450786191, −1.11244569338940477324935281031, 0,
1.11244569338940477324935281031, 1.79189098456849711197450786191, 2.67539943625420951049270226971, 3.44449310979811290848798183696, 3.97468086120425923627726323610, 4.57160537934393565707213291122, 4.76158685629415364500613814436, 5.36701953076705401568809187494, 5.79738003881247977630639495666, 6.30044424957163342752387110170, 6.64441444704426775177764403722, 7.57268670837969735844538288727, 7.69800230033996657873823491182, 8.127471297869059558051546284898
