L(s) = 1 | − 3-s + 7-s − 2·9-s + 6·11-s + 3·13-s − 21-s + 23-s − 5·25-s + 5·27-s + 3·29-s − 7·31-s − 6·33-s − 8·37-s − 3·39-s − 11·41-s − 4·43-s + 47-s + 49-s − 4·53-s − 12·59-s + 6·61-s − 2·63-s − 12·67-s − 69-s − 5·71-s + 15·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s + 0.832·13-s − 0.218·21-s + 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s − 1.25·31-s − 1.04·33-s − 1.31·37-s − 0.480·39-s − 1.71·41-s − 0.609·43-s + 0.145·47-s + 1/7·49-s − 0.549·53-s − 1.56·59-s + 0.768·61-s − 0.251·63-s − 1.46·67-s − 0.120·69-s − 0.593·71-s + 1.75·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.80835305113360, −16.66587056246160, −15.66983247643006, −15.25931902940550, −14.45720536121246, −14.05659894524625, −13.63125940506927, −12.72707853213356, −11.99807067321074, −11.71632357317441, −11.19155123981359, −10.60922132016348, −9.875421486732918, −9.019228761680257, −8.744261333362011, −8.042019150964037, −7.096270072234887, −6.533002590536844, −5.997000916113788, −5.331277079550149, −4.580131821802789, −3.725409319125754, −3.245729740580509, −1.869001162576256, −1.276997807937623, 0,
1.276997807937623, 1.869001162576256, 3.245729740580509, 3.725409319125754, 4.580131821802789, 5.331277079550149, 5.997000916113788, 6.533002590536844, 7.096270072234887, 8.042019150964037, 8.744261333362011, 9.019228761680257, 9.875421486732918, 10.60922132016348, 11.19155123981359, 11.71632357317441, 11.99807067321074, 12.72707853213356, 13.63125940506927, 14.05659894524625, 14.45720536121246, 15.25931902940550, 15.66983247643006, 16.66587056246160, 16.80835305113360