| L(s) = 1 | − 3-s − 2.16·5-s − 7-s + 9-s − 3·11-s − 0.162·13-s + 2.16·15-s − 17-s + 1.83·19-s + 21-s − 7.32·23-s − 0.324·25-s − 27-s + 10.3·29-s − 1.16·31-s + 3·33-s + 2.16·35-s − 9.16·37-s + 0.162·39-s − 3.83·41-s − 9.32·43-s − 2.16·45-s − 5.16·47-s + 49-s + 51-s + 9.48·53-s + 6.48·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.966·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s − 0.0450·13-s + 0.558·15-s − 0.242·17-s + 0.421·19-s + 0.218·21-s − 1.52·23-s − 0.0649·25-s − 0.192·27-s + 1.91·29-s − 0.208·31-s + 0.522·33-s + 0.365·35-s − 1.50·37-s + 0.0259·39-s − 0.599·41-s − 1.42·43-s − 0.322·45-s − 0.752·47-s + 0.142·49-s + 0.140·51-s + 1.30·53-s + 0.874·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5092902909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5092902909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2.16T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 0.162T + 13T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 7.16T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.220094241865914411058141860379, −7.34744559564930105584374954016, −6.77212343936618090169346311650, −5.98871350518501796244378121884, −5.18938886017655228095240878419, −4.52788835494880625515537032491, −3.70110526043721509999793455388, −2.95321312026719671999130632615, −1.79962660521271483312640355686, −0.37762336384834728616715225670,
0.37762336384834728616715225670, 1.79962660521271483312640355686, 2.95321312026719671999130632615, 3.70110526043721509999793455388, 4.52788835494880625515537032491, 5.18938886017655228095240878419, 5.98871350518501796244378121884, 6.77212343936618090169346311650, 7.34744559564930105584374954016, 8.220094241865914411058141860379
