| L(s) = 1 | + 2.24e13·2-s − 1.14e26·4-s + 1.81e31·5-s − 2.76e37·7-s − 1.64e40·8-s + 4.07e44·10-s − 2.89e46·11-s − 2.93e49·13-s − 6.20e50·14-s − 2.99e53·16-s − 3.53e54·17-s − 2.73e56·19-s − 2.07e57·20-s − 6.49e59·22-s − 5.99e60·23-s + 1.66e62·25-s − 6.58e62·26-s + 3.15e63·28-s − 2.05e65·29-s − 1.59e66·31-s + 3.47e66·32-s − 7.94e67·34-s − 5.00e68·35-s + 1.13e69·37-s − 6.13e69·38-s − 2.98e71·40-s − 8.14e70·41-s + ⋯ |
| L(s) = 1 | + 0.902·2-s − 0.184·4-s + 1.42·5-s − 0.682·7-s − 1.06·8-s + 1.28·10-s − 1.31·11-s − 0.787·13-s − 0.616·14-s − 0.781·16-s − 0.621·17-s − 0.340·19-s − 0.263·20-s − 1.18·22-s − 1.51·23-s + 1.03·25-s − 0.711·26-s + 0.126·28-s − 1.72·29-s − 0.685·31-s + 0.364·32-s − 0.561·34-s − 0.973·35-s + 0.186·37-s − 0.307·38-s − 1.52·40-s − 0.138·41-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
| \(L(45)\) |
\(\approx\) |
\(1.113398585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113398585\) |
| \(L(\frac{91}{2})\) |
|
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 | \( 1 - 2.24e13T + 6.18e26T^{2} \) |
| 5 | \( 1 - 1.81e31T + 1.61e62T^{2} \) |
| 7 | \( 1 + 2.76e37T + 1.63e75T^{2} \) |
| 11 | \( 1 + 2.89e46T + 4.83e92T^{2} \) |
| 13 | \( 1 + 2.93e49T + 1.38e99T^{2} \) |
| 17 | \( 1 + 3.53e54T + 3.23e109T^{2} \) |
| 19 | \( 1 + 2.73e56T + 6.44e113T^{2} \) |
| 23 | \( 1 + 5.99e60T + 1.56e121T^{2} \) |
| 29 | \( 1 + 2.05e65T + 1.42e130T^{2} \) |
| 31 | \( 1 + 1.59e66T + 5.38e132T^{2} \) |
| 37 | \( 1 - 1.13e69T + 3.71e139T^{2} \) |
| 41 | \( 1 + 8.14e70T + 3.44e143T^{2} \) |
| 43 | \( 1 - 3.18e72T + 2.39e145T^{2} \) |
| 47 | \( 1 - 3.19e74T + 6.55e148T^{2} \) |
| 53 | \( 1 - 1.04e77T + 2.88e153T^{2} \) |
| 59 | \( 1 + 8.65e78T + 4.03e157T^{2} \) |
| 61 | \( 1 + 3.87e79T + 7.84e158T^{2} \) |
| 67 | \( 1 - 1.41e81T + 3.31e162T^{2} \) |
| 71 | \( 1 + 1.52e82T + 5.78e164T^{2} \) |
| 73 | \( 1 + 5.28e82T + 6.85e165T^{2} \) |
| 79 | \( 1 - 4.95e84T + 7.74e168T^{2} \) |
| 83 | \( 1 + 5.30e84T + 6.27e170T^{2} \) |
| 89 | \( 1 - 6.27e86T + 3.13e173T^{2} \) |
| 97 | \( 1 - 2.15e88T + 6.64e176T^{2} \) |
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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
−9.727537725684921260401443226245, −9.006117366240243279794846521116, −7.50706651888815622748102262573, −6.13606520911848624973917756673, −5.67147407786051266680588693489, −4.78013124614948202062915193824, −3.66513405428049908543623868267, −2.51122215625549974022614204889, −2.04956959899829664214065919916, −0.29879552897640538130313128856,
0.29879552897640538130313128856, 2.04956959899829664214065919916, 2.51122215625549974022614204889, 3.66513405428049908543623868267, 4.78013124614948202062915193824, 5.67147407786051266680588693489, 6.13606520911848624973917756673, 7.50706651888815622748102262573, 9.006117366240243279794846521116, 9.727537725684921260401443226245
