| L(s) = 1 | − 1.19·2-s + 3.04·3-s − 0.581·4-s − 3.62·6-s + 0.609·7-s + 3.07·8-s + 6.28·9-s + 4.48·11-s − 1.77·12-s + 4.43·13-s − 0.725·14-s − 2.49·16-s − 2.90·17-s − 7.48·18-s + 1.85·21-s − 5.34·22-s + 2.84·23-s + 9.36·24-s − 5.28·26-s + 10.0·27-s − 0.354·28-s − 1.11·29-s + 6.22·31-s − 3.17·32-s + 13.6·33-s + 3.45·34-s − 3.65·36-s + ⋯ |
| L(s) = 1 | − 0.842·2-s + 1.75·3-s − 0.290·4-s − 1.48·6-s + 0.230·7-s + 1.08·8-s + 2.09·9-s + 1.35·11-s − 0.511·12-s + 1.23·13-s − 0.193·14-s − 0.624·16-s − 0.704·17-s − 1.76·18-s + 0.405·21-s − 1.13·22-s + 0.593·23-s + 1.91·24-s − 1.03·26-s + 1.92·27-s − 0.0669·28-s − 0.207·29-s + 1.11·31-s − 0.561·32-s + 2.37·33-s + 0.592·34-s − 0.609·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.163653420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.163653420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 - 0.609T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 23 | \( 1 - 2.84T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 - 5.88T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−7.993762183400419301026762011932, −7.34037530852256859991558933046, −6.73464565907127047298610925009, −5.82935994108045707204639872117, −4.42633147690899610238170169119, −4.22016515258556046400663224144, −3.43748432181038534560541903929, −2.53586442675143376565979766784, −1.56755634933443402349990924665, −1.03175658801345090660419141278,
1.03175658801345090660419141278, 1.56755634933443402349990924665, 2.53586442675143376565979766784, 3.43748432181038534560541903929, 4.22016515258556046400663224144, 4.42633147690899610238170169119, 5.82935994108045707204639872117, 6.73464565907127047298610925009, 7.34037530852256859991558933046, 7.993762183400419301026762011932
