| L(s) = 1 | + 2.25·3-s + 4.22·7-s + 2.08·9-s + 5.13·11-s − 3.16·13-s + 6.48·17-s + 19-s + 9.53·21-s + 7.56·23-s − 2.05·27-s + 0.832·29-s + 4.51·31-s + 11.5·33-s − 0.137·37-s − 7.14·39-s − 11.6·41-s − 2.51·43-s − 5.96·47-s + 10.8·49-s + 14.6·51-s − 0.225·53-s + 2.25·57-s − 5.39·59-s + 14.4·61-s + 8.82·63-s − 4.11·67-s + 17.0·69-s + ⋯ |
| L(s) = 1 | + 1.30·3-s + 1.59·7-s + 0.695·9-s + 1.54·11-s − 0.878·13-s + 1.57·17-s + 0.229·19-s + 2.07·21-s + 1.57·23-s − 0.395·27-s + 0.154·29-s + 0.810·31-s + 2.01·33-s − 0.0226·37-s − 1.14·39-s − 1.81·41-s − 0.382·43-s − 0.869·47-s + 1.55·49-s + 2.04·51-s − 0.0309·53-s + 0.298·57-s − 0.702·59-s + 1.85·61-s + 1.11·63-s − 0.502·67-s + 2.05·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.902505915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.902505915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 - 0.832T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 + 0.137T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−8.076048685670878083538737899067, −7.28462674437544231024862277365, −6.83698547050169396648116876920, −5.61095502502052187121003829737, −4.94688843803910771983976290278, −4.27237880741656956635406376022, −3.38942683707598686297615534334, −2.79880507112067974630428737314, −1.67905409083544611693758489194, −1.22468851395311659801481795008,
1.22468851395311659801481795008, 1.67905409083544611693758489194, 2.79880507112067974630428737314, 3.38942683707598686297615534334, 4.27237880741656956635406376022, 4.94688843803910771983976290278, 5.61095502502052187121003829737, 6.83698547050169396648116876920, 7.28462674437544231024862277365, 8.076048685670878083538737899067
