| L(s) = 1 | + 3-s − 2.19·5-s − 1.02·7-s + 9-s − 1.96·11-s + 6.05·13-s − 2.19·15-s + 3.27·17-s + 4.49·19-s − 1.02·21-s − 2.00·23-s − 0.189·25-s + 27-s − 9.65·29-s + 0.0109·31-s − 1.96·33-s + 2.24·35-s − 6.52·37-s + 6.05·39-s + 7.05·41-s + 2.48·43-s − 2.19·45-s + 1.91·47-s − 5.95·49-s + 3.27·51-s + 4.44·53-s + 4.29·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.980·5-s − 0.387·7-s + 0.333·9-s − 0.591·11-s + 1.67·13-s − 0.566·15-s + 0.794·17-s + 1.03·19-s − 0.223·21-s − 0.418·23-s − 0.0378·25-s + 0.192·27-s − 1.79·29-s + 0.00197·31-s − 0.341·33-s + 0.379·35-s − 1.07·37-s + 0.969·39-s + 1.10·41-s + 0.378·43-s − 0.326·45-s + 0.278·47-s − 0.850·49-s + 0.458·51-s + 0.610·53-s + 0.579·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926272849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926272849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 6.05T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 0.0109T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 + 0.802T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.042884342638363110000409139669, −7.54519308080827435449020787543, −6.86237217827326105048128922126, −5.83663211093723504683727464541, −5.33251881921163272918660010913, −4.04751181394230896580233023061, −3.66958296245177609470993093474, −3.05179231874652022006184629221, −1.85300857530382458238038126078, −0.71598170268704531368821598633,
0.71598170268704531368821598633, 1.85300857530382458238038126078, 3.05179231874652022006184629221, 3.66958296245177609470993093474, 4.04751181394230896580233023061, 5.33251881921163272918660010913, 5.83663211093723504683727464541, 6.86237217827326105048128922126, 7.54519308080827435449020787543, 8.042884342638363110000409139669
