| L(s) = 1 | − 642.·2-s − 5.85e7·3-s − 8.58e9·4-s + 3.76e10·6-s + 1.03e14·7-s + 1.10e13·8-s − 2.13e15·9-s − 4.04e16·11-s + 5.02e17·12-s − 3.61e18·13-s − 6.68e16·14-s + 7.37e19·16-s + 2.12e20·17-s + 1.37e18·18-s − 6.69e20·19-s − 6.08e21·21-s + 2.60e19·22-s + 4.58e21·23-s − 6.46e20·24-s + 2.32e21·26-s + 4.50e23·27-s − 8.92e23·28-s − 2.08e24·29-s − 2.71e24·31-s − 1.42e23·32-s + 2.36e24·33-s − 1.36e23·34-s + ⋯ |
| L(s) = 1 | − 0.00693·2-s − 0.785·3-s − 0.999·4-s + 0.00544·6-s + 1.18·7-s + 0.0138·8-s − 0.383·9-s − 0.265·11-s + 0.784·12-s − 1.50·13-s − 0.00819·14-s + 0.999·16-s + 1.05·17-s + 0.00266·18-s − 0.532·19-s − 0.927·21-s + 0.00184·22-s + 0.156·23-s − 0.0108·24-s + 0.0104·26-s + 1.08·27-s − 1.18·28-s − 1.54·29-s − 0.671·31-s − 0.0208·32-s + 0.208·33-s − 0.00734·34-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 | \( 1 + 642.T + 8.58e9T^{2} \) |
| 3 | \( 1 + 5.85e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.03e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 4.04e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.61e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.12e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 6.69e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 4.58e21T + 8.65e44T^{2} \) |
| 29 | \( 1 + 2.08e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.71e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 2.98e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 2.01e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 9.42e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 4.81e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 1.34e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 3.17e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.25e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 3.16e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 5.91e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.98e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 4.21e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 2.51e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.35e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.72e32T + 3.65e65T^{2} \) |
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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
−10.77734791389991748492346905102, −9.616374187406537970395939408140, −8.363255512422673743976268135643, −7.38289660616790005701451232368, −5.50361457115652784985428911367, −5.14934957206202974889578673844, −3.96275317327208997829003945481, −2.32931912818512413332700952144, −0.917260154756859688639554706363, 0,
0.917260154756859688639554706363, 2.32931912818512413332700952144, 3.96275317327208997829003945481, 5.14934957206202974889578673844, 5.50361457115652784985428911367, 7.38289660616790005701451232368, 8.363255512422673743976268135643, 9.616374187406537970395939408140, 10.77734791389991748492346905102
