| L(s) = 1 | − 1.92e13·2-s − 2.48e26·4-s + 3.02e30·5-s + 1.70e37·7-s + 1.66e40·8-s − 5.82e43·10-s − 1.93e46·11-s + 1.74e49·13-s − 3.28e50·14-s − 1.67e53·16-s + 6.51e54·17-s − 9.59e56·19-s − 7.53e56·20-s + 3.71e59·22-s + 9.89e59·23-s − 1.52e62·25-s − 3.35e62·26-s − 4.24e63·28-s + 1.35e65·29-s + 1.59e66·31-s − 7.11e66·32-s − 1.25e68·34-s + 5.16e67·35-s − 2.47e68·37-s + 1.84e70·38-s + 5.05e70·40-s − 4.58e71·41-s + ⋯ |
| L(s) = 1 | − 0.773·2-s − 0.401·4-s + 0.238·5-s + 0.421·7-s + 1.08·8-s − 0.184·10-s − 0.878·11-s + 0.468·13-s − 0.326·14-s − 0.436·16-s + 1.14·17-s − 1.19·19-s − 0.0957·20-s + 0.679·22-s + 0.250·23-s − 0.943·25-s − 0.362·26-s − 0.169·28-s + 1.13·29-s + 0.686·31-s − 0.746·32-s − 0.885·34-s + 0.100·35-s − 0.0406·37-s + 0.925·38-s + 0.258·40-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(90-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+89/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(45)\) |
\(\approx\) |
\(1.186270911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186270911\) |
| \(L(\frac{91}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.92e13T + 6.18e26T^{2} \) |
| 5 | \( 1 - 3.02e30T + 1.61e62T^{2} \) |
| 7 | \( 1 - 1.70e37T + 1.63e75T^{2} \) |
| 11 | \( 1 + 1.93e46T + 4.83e92T^{2} \) |
| 13 | \( 1 - 1.74e49T + 1.38e99T^{2} \) |
| 17 | \( 1 - 6.51e54T + 3.23e109T^{2} \) |
| 19 | \( 1 + 9.59e56T + 6.44e113T^{2} \) |
| 23 | \( 1 - 9.89e59T + 1.56e121T^{2} \) |
| 29 | \( 1 - 1.35e65T + 1.42e130T^{2} \) |
| 31 | \( 1 - 1.59e66T + 5.38e132T^{2} \) |
| 37 | \( 1 + 2.47e68T + 3.71e139T^{2} \) |
| 41 | \( 1 + 4.58e71T + 3.44e143T^{2} \) |
| 43 | \( 1 + 7.07e72T + 2.39e145T^{2} \) |
| 47 | \( 1 + 1.32e74T + 6.55e148T^{2} \) |
| 53 | \( 1 - 9.22e76T + 2.88e153T^{2} \) |
| 59 | \( 1 - 8.06e78T + 4.03e157T^{2} \) |
| 61 | \( 1 - 4.49e79T + 7.84e158T^{2} \) |
| 67 | \( 1 - 1.54e81T + 3.31e162T^{2} \) |
| 71 | \( 1 - 2.24e82T + 5.78e164T^{2} \) |
| 73 | \( 1 + 1.49e83T + 6.85e165T^{2} \) |
| 79 | \( 1 - 3.12e84T + 7.74e168T^{2} \) |
| 83 | \( 1 + 3.77e85T + 6.27e170T^{2} \) |
| 89 | \( 1 + 2.53e85T + 3.13e173T^{2} \) |
| 97 | \( 1 - 3.77e87T + 6.64e176T^{2} \) |
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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
−9.907358323011606650284440588971, −8.431727880281079334566452238613, −8.174294589574766799741994215542, −6.87072593072694321461726806398, −5.55494205159428515065587284914, −4.71473977250864164811375371938, −3.61815325744157683914707474086, −2.30482094325129702670551845966, −1.35484589193307368268871716173, −0.48017640817555089281072490913,
0.48017640817555089281072490913, 1.35484589193307368268871716173, 2.30482094325129702670551845966, 3.61815325744157683914707474086, 4.71473977250864164811375371938, 5.55494205159428515065587284914, 6.87072593072694321461726806398, 8.174294589574766799741994215542, 8.431727880281079334566452238613, 9.907358323011606650284440588971
