| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 9-s + 2·10-s − 3·11-s − 7·13-s − 14-s + 16-s − 18-s − 2·20-s + 3·22-s + 2·25-s + 7·26-s + 28-s − 7·31-s − 32-s − 2·35-s + 36-s + 2·40-s − 8·43-s − 3·44-s − 2·45-s − 6·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 1.94·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.447·20-s + 0.639·22-s + 2/5·25-s + 1.37·26-s + 0.188·28-s − 1.25·31-s − 0.176·32-s − 0.338·35-s + 1/6·36-s + 0.316·40-s − 1.21·43-s − 0.452·44-s − 0.298·45-s − 6/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.20919657149063977349669606384, −9.728844590354016560148994714170, −9.218066556667966046804252131106, −8.563299507423901263052802000050, −7.954435151686005823725761040893, −7.61897456432862457817693214764, −7.22483825910355129771116564000, −6.65991795355580065091763579290, −5.71032897742396072140555396904, −4.97213677381490305674521003367, −4.62497032074201372261197328357, −3.58997653594847791536925089743, −2.77947941066286340719912918967, −1.87558216817968593370036781378, 0,
1.87558216817968593370036781378, 2.77947941066286340719912918967, 3.58997653594847791536925089743, 4.62497032074201372261197328357, 4.97213677381490305674521003367, 5.71032897742396072140555396904, 6.65991795355580065091763579290, 7.22483825910355129771116564000, 7.61897456432862457817693214764, 7.954435151686005823725761040893, 8.563299507423901263052802000050, 9.218066556667966046804252131106, 9.728844590354016560148994714170, 10.20919657149063977349669606384
