| L(s) = 1 | + 2.80e12·2-s − 9.69e19·3-s − 1.77e24·4-s + 1.10e29·5-s − 2.72e32·6-s − 5.50e34·7-s − 3.21e37·8-s + 5.41e39·9-s + 3.11e41·10-s − 2.22e43·11-s + 1.72e44·12-s − 6.42e45·13-s − 1.54e47·14-s − 1.07e49·15-s − 7.32e49·16-s − 1.88e51·17-s + 1.52e52·18-s + 1.37e53·19-s − 1.96e53·20-s + 5.34e54·21-s − 6.24e55·22-s + 6.47e56·23-s + 3.11e57·24-s + 1.96e57·25-s − 1.80e58·26-s − 1.38e59·27-s + 9.78e58·28-s + ⋯ |
| L(s) = 1 | + 0.903·2-s − 1.53·3-s − 0.183·4-s + 1.09·5-s − 1.38·6-s − 0.467·7-s − 1.06·8-s + 1.35·9-s + 0.985·10-s − 1.34·11-s + 0.281·12-s − 0.379·13-s − 0.422·14-s − 1.67·15-s − 0.782·16-s − 1.62·17-s + 1.22·18-s + 1.17·19-s − 0.200·20-s + 0.717·21-s − 1.21·22-s + 1.99·23-s + 1.64·24-s + 0.189·25-s − 0.343·26-s − 0.548·27-s + 0.0857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(84-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+83/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(42)\) |
\(\approx\) |
\(1.232685589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232685589\) |
| \(L(\frac{85}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.80e12T + 9.67e24T^{2} \) |
| 3 | \( 1 + 9.69e19T + 3.99e39T^{2} \) |
| 5 | \( 1 - 1.10e29T + 1.03e58T^{2} \) |
| 7 | \( 1 + 5.50e34T + 1.39e70T^{2} \) |
| 11 | \( 1 + 2.22e43T + 2.72e86T^{2} \) |
| 13 | \( 1 + 6.42e45T + 2.86e92T^{2} \) |
| 17 | \( 1 + 1.88e51T + 1.34e102T^{2} \) |
| 19 | \( 1 - 1.37e53T + 1.36e106T^{2} \) |
| 23 | \( 1 - 6.47e56T + 1.05e113T^{2} \) |
| 29 | \( 1 - 6.73e59T + 2.39e121T^{2} \) |
| 31 | \( 1 - 4.12e61T + 6.06e123T^{2} \) |
| 37 | \( 1 + 4.23e63T + 1.44e130T^{2} \) |
| 41 | \( 1 - 1.30e67T + 7.26e133T^{2} \) |
| 43 | \( 1 + 3.87e67T + 3.78e135T^{2} \) |
| 47 | \( 1 - 2.21e69T + 6.08e138T^{2} \) |
| 53 | \( 1 + 3.92e70T + 1.30e143T^{2} \) |
| 59 | \( 1 + 3.16e73T + 9.56e146T^{2} \) |
| 61 | \( 1 - 5.48e73T + 1.52e148T^{2} \) |
| 67 | \( 1 - 6.60e75T + 3.66e151T^{2} \) |
| 71 | \( 1 - 7.10e76T + 4.51e153T^{2} \) |
| 73 | \( 1 - 9.86e76T + 4.52e154T^{2} \) |
| 79 | \( 1 - 1.00e79T + 3.18e157T^{2} \) |
| 83 | \( 1 + 4.07e79T + 1.92e159T^{2} \) |
| 89 | \( 1 + 2.55e80T + 6.30e161T^{2} \) |
| 97 | \( 1 + 7.47e81T + 7.98e164T^{2} \) |
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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
−15.64502781389639151281926931372, −13.52298877814259844700425291550, −12.67200889532769687941505314713, −11.00256910527710908393911634752, −9.514461708203551562173523278820, −6.63409522856767612855979762336, −5.49885798871510605018542993899, −4.82198524221240653246091862956, −2.68510262677656696749332295313, −0.58892235545161152057114558656,
0.58892235545161152057114558656, 2.68510262677656696749332295313, 4.82198524221240653246091862956, 5.49885798871510605018542993899, 6.63409522856767612855979762336, 9.514461708203551562173523278820, 11.00256910527710908393911634752, 12.67200889532769687941505314713, 13.52298877814259844700425291550, 15.64502781389639151281926931372
