| L(s) = 1 | − 3.07e11·2-s + 5.70e22·4-s + 2.50e26·5-s + 8.88e31·7-s − 5.92e33·8-s − 7.70e37·10-s − 5.59e38·11-s − 5.18e41·13-s − 2.73e43·14-s − 3.30e44·16-s − 9.59e45·17-s − 6.50e47·19-s + 1.42e49·20-s + 1.72e50·22-s + 1.30e51·23-s + 3.61e52·25-s + 1.59e53·26-s + 5.06e54·28-s − 1.20e55·29-s − 2.86e55·31-s + 3.25e56·32-s + 2.95e57·34-s + 2.22e58·35-s + 1.23e58·37-s + 2.00e59·38-s − 1.48e60·40-s − 4.04e60·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 1.53·5-s + 1.80·7-s − 0.806·8-s − 2.43·10-s − 0.496·11-s − 0.874·13-s − 2.86·14-s − 0.231·16-s − 0.692·17-s − 0.724·19-s + 2.32·20-s + 0.786·22-s + 1.12·23-s + 1.36·25-s + 1.38·26-s + 2.73·28-s − 1.74·29-s − 0.339·31-s + 1.17·32-s + 1.09·34-s + 2.78·35-s + 0.192·37-s + 1.14·38-s − 1.24·40-s − 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{77}{2})\) |
|
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 | \( 1 + 3.07e11T + 3.77e22T^{2} \) |
| 5 | \( 1 - 2.50e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 8.88e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 5.59e38T + 1.27e78T^{2} \) |
| 13 | \( 1 + 5.18e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 9.59e45T + 1.92e92T^{2} \) |
| 19 | \( 1 + 6.50e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 1.30e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 1.20e55T + 4.78e109T^{2} \) |
| 31 | \( 1 + 2.86e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.23e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 4.04e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 2.97e61T + 3.23e122T^{2} \) |
| 47 | \( 1 + 7.18e61T + 2.55e125T^{2} \) |
| 53 | \( 1 + 7.66e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 1.60e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.51e67T + 7.93e133T^{2} \) |
| 67 | \( 1 + 4.17e67T + 9.02e136T^{2} \) |
| 71 | \( 1 - 1.82e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 2.79e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.40e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 8.54e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.72e72T + 1.60e146T^{2} \) |
| 97 | \( 1 + 4.41e74T + 1.01e149T^{2} \) |
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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
−9.596492734878186435832730695468, −8.831675327087711884786870149732, −7.85214418848932956088697056652, −6.91052511630168298808630296720, −5.51798924999615794662623030565, −4.67695443967387363639163456212, −2.39666319183635333388756425632, −1.94560793183230443610115898962, −1.21883647426628872474832637816, 0,
1.21883647426628872474832637816, 1.94560793183230443610115898962, 2.39666319183635333388756425632, 4.67695443967387363639163456212, 5.51798924999615794662623030565, 6.91052511630168298808630296720, 7.85214418848932956088697056652, 8.831675327087711884786870149732, 9.596492734878186435832730695468
