| L(s) = 1 | + 7.97e22·3-s + 7.92e28·4-s + 6.06e40·7-s + 6.36e45·9-s + 6.31e51·12-s − 1.60e53·13-s + 6.27e57·16-s + 1.95e60·19-s + 4.84e63·21-s + 1.26e67·25-s + 5.07e68·27-s + 4.80e69·28-s − 5.81e71·31-s + 5.04e74·36-s − 1.73e74·37-s − 1.28e76·39-s + 5.04e78·43-s + 5.00e80·48-s + 2.33e81·49-s − 1.27e82·52-s + 1.55e83·57-s + 9.84e85·61-s + 3.86e86·63-s + 4.97e86·64-s − 4.33e86·67-s − 5.21e89·73-s + 1.00e90·75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 1.65·7-s + 9-s + 12-s − 0.544·13-s + 16-s + 0.0813·19-s + 1.65·21-s + 25-s + 27-s + 1.65·28-s − 1.51·31-s + 36-s − 0.0921·37-s − 0.544·39-s + 1.97·43-s + 48-s + 1.73·49-s − 0.544·52-s + 0.0813·57-s + 1.98·61-s + 1.65·63-s + 64-s − 0.0966·67-s − 1.89·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(97-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+48) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{97}{2})\) |
\(\approx\) |
\(6.380626807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.380626807\) |
| \(L(49)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{48} T \) |
| good | 2 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 5 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 7 | \( 1 - \)\(60\!\cdots\!02\)\( T + p^{96} T^{2} \) |
| 11 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 13 | \( 1 + \)\(16\!\cdots\!58\)\( T + p^{96} T^{2} \) |
| 17 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 19 | \( 1 - \)\(19\!\cdots\!82\)\( T + p^{96} T^{2} \) |
| 23 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 29 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 31 | \( 1 + \)\(58\!\cdots\!18\)\( T + p^{96} T^{2} \) |
| 37 | \( 1 + \)\(17\!\cdots\!58\)\( T + p^{96} T^{2} \) |
| 41 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 43 | \( 1 - \)\(50\!\cdots\!02\)\( T + p^{96} T^{2} \) |
| 47 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 53 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 59 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 61 | \( 1 - \)\(98\!\cdots\!62\)\( T + p^{96} T^{2} \) |
| 67 | \( 1 + \)\(43\!\cdots\!18\)\( T + p^{96} T^{2} \) |
| 71 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 73 | \( 1 + \)\(52\!\cdots\!38\)\( T + p^{96} T^{2} \) |
| 79 | \( 1 + \)\(23\!\cdots\!78\)\( T + p^{96} T^{2} \) |
| 83 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 89 | \( ( 1 - p^{48} T )( 1 + p^{48} T ) \) |
| 97 | \( 1 - \)\(40\!\cdots\!22\)\( T + p^{96} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42391333606611090698477472995, −10.42580233344408102010434196544, −8.880838946169448676690429343578, −7.78964077457976969097592514276, −7.09949566179347181953906376439, −5.37774841661965049495155237774, −4.15002962446715121384281721160, −2.77459399880729475533907056866, −1.94417729537391892507251115011, −1.12876936078044295985257025831,
1.12876936078044295985257025831, 1.94417729537391892507251115011, 2.77459399880729475533907056866, 4.15002962446715121384281721160, 5.37774841661965049495155237774, 7.09949566179347181953906376439, 7.78964077457976969097592514276, 8.880838946169448676690429343578, 10.42580233344408102010434196544, 11.42391333606611090698477472995
