L(s) = 1 | + 2-s + 4-s − 4·5-s + 8-s − 6·9-s − 4·10-s + 4·13-s + 16-s − 12·17-s − 6·18-s − 4·20-s + 2·25-s + 4·26-s + 4·29-s + 32-s − 12·34-s − 6·36-s + 20·37-s − 4·40-s − 12·41-s + 24·45-s − 14·49-s + 2·50-s + 4·52-s − 12·53-s + 4·58-s − 12·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 2·9-s − 1.26·10-s + 1.10·13-s + 1/4·16-s − 2.91·17-s − 1.41·18-s − 0.894·20-s + 2/5·25-s + 0.784·26-s + 0.742·29-s + 0.176·32-s − 2.05·34-s − 36-s + 3.28·37-s − 0.632·40-s − 1.87·41-s + 3.57·45-s − 2·49-s + 0.282·50-s + 0.554·52-s − 1.64·53-s + 0.525·58-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72587641522723560036260231629, −9.557473081917962507344751915689, −9.056498267458590117851162621111, −8.371776216834959175000943223311, −8.156371355658038539653222782109, −7.76857292337562945056886541715, −6.72038595262299226278216985203, −6.32858579420662158347624019611, −5.94945525164915580359156684875, −4.74203912193695697715499891246, −4.55787579523055807179904019114, −3.68623363482399645957944200957, −3.16188554299090604324575557253, −2.29075936305574985962112760644, 0,
2.29075936305574985962112760644, 3.16188554299090604324575557253, 3.68623363482399645957944200957, 4.55787579523055807179904019114, 4.74203912193695697715499891246, 5.94945525164915580359156684875, 6.32858579420662158347624019611, 6.72038595262299226278216985203, 7.76857292337562945056886541715, 8.156371355658038539653222782109, 8.371776216834959175000943223311, 9.056498267458590117851162621111, 9.557473081917962507344751915689, 10.72587641522723560036260231629