| L(s) = 1 | + 1.53e5·2-s + 7.94e7·3-s + 1.51e10·4-s + 1.22e13·6-s − 4.85e13·7-s + 1.00e15·8-s + 7.57e14·9-s − 1.47e17·11-s + 1.20e18·12-s − 2.08e18·13-s − 7.46e18·14-s + 2.50e19·16-s + 2.59e20·17-s + 1.16e20·18-s − 1.93e21·19-s − 3.85e21·21-s − 2.27e22·22-s + 2.85e22·23-s + 7.99e22·24-s − 3.21e23·26-s − 3.81e23·27-s − 7.33e23·28-s − 3.23e22·29-s − 2.36e24·31-s − 4.78e24·32-s − 1.17e25·33-s + 3.99e25·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.06·3-s + 1.76·4-s + 1.77·6-s − 0.551·7-s + 1.26·8-s + 0.136·9-s − 0.968·11-s + 1.87·12-s − 0.869·13-s − 0.916·14-s + 0.339·16-s + 1.29·17-s + 0.226·18-s − 1.53·19-s − 0.588·21-s − 1.60·22-s + 0.971·23-s + 1.34·24-s − 1.44·26-s − 0.920·27-s − 0.971·28-s − 0.0239·29-s − 0.583·31-s − 0.700·32-s − 1.03·33-s + 2.14·34-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 | \( 1 - 1.53e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 7.94e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 4.85e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 1.47e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.08e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.59e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.93e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 2.85e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 3.23e22T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.36e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 3.95e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 1.88e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.41e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 5.67e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 3.48e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.09e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 4.19e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.51e30T + 1.82e60T^{2} \) |
| 71 | \( 1 - 1.38e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 6.30e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.34e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 4.79e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.97e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 6.00e32T + 3.65e65T^{2} \) |
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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
−10.98098117620854137337939083861, −9.608350857962857283310936111315, −8.168443315704914938465020042692, −7.02010427093622091738493280699, −5.74275325802993539363501902594, −4.74414223396998942727161508504, −3.48520619691067539352581086541, −2.86346031380182196397210204914, −2.01248929553888847029773490303, 0,
2.01248929553888847029773490303, 2.86346031380182196397210204914, 3.48520619691067539352581086541, 4.74414223396998942727161508504, 5.74275325802993539363501902594, 7.02010427093622091738493280699, 8.168443315704914938465020042692, 9.608350857962857283310936111315, 10.98098117620854137337939083861
