| L(s) = 1 | + 5.61e4·2-s − 5.48e7·3-s − 5.44e9·4-s − 3.07e12·6-s − 7.38e13·7-s − 7.87e14·8-s − 2.54e15·9-s + 3.54e16·11-s + 2.98e17·12-s + 1.27e18·13-s − 4.14e18·14-s + 2.54e18·16-s − 1.95e20·17-s − 1.43e20·18-s + 9.67e20·19-s + 4.05e21·21-s + 1.98e21·22-s + 3.06e22·23-s + 4.31e22·24-s + 7.14e22·26-s + 4.44e23·27-s + 4.01e23·28-s − 1.16e23·29-s − 3.89e23·31-s + 6.90e24·32-s − 1.94e24·33-s − 1.09e25·34-s + ⋯ |
| L(s) = 1 | + 0.605·2-s − 0.735·3-s − 0.633·4-s − 0.445·6-s − 0.839·7-s − 0.989·8-s − 0.458·9-s + 0.232·11-s + 0.466·12-s + 0.530·13-s − 0.508·14-s + 0.0345·16-s − 0.974·17-s − 0.277·18-s + 0.769·19-s + 0.618·21-s + 0.140·22-s + 1.04·23-s + 0.727·24-s + 0.321·26-s + 1.07·27-s + 0.532·28-s − 0.0862·29-s − 0.0961·31-s + 1.00·32-s − 0.171·33-s − 0.589·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 5.61e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 5.48e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 7.38e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 3.54e16T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.27e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 1.95e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 9.67e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.06e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.16e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 3.89e23T + 1.64e49T^{2} \) |
| 37 | \( 1 - 2.73e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 7.27e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.74e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.14e26T + 1.51e55T^{2} \) |
| 53 | \( 1 - 1.96e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 2.54e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.81e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 8.18e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.59e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.19e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 3.90e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 4.98e30T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.23e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.97e32T + 3.65e65T^{2} \) |
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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
−10.93936119236840008009074173380, −9.512598931790211816468646033602, −8.607758073138372182513348305804, −6.78449132661808434344696407140, −5.86897457536906795953650918211, −4.96673626506549544095369568525, −3.77107204593090400632236068788, −2.78780543234006631790662494330, −0.903696824078740382983761642345, 0,
0.903696824078740382983761642345, 2.78780543234006631790662494330, 3.77107204593090400632236068788, 4.96673626506549544095369568525, 5.86897457536906795953650918211, 6.78449132661808434344696407140, 8.607758073138372182513348305804, 9.512598931790211816468646033602, 10.93936119236840008009074173380
