| L(s) = 1 | − 0.635·2-s − 1.02·3-s − 1.59·4-s + 5-s + 0.654·6-s + 2.28·8-s − 1.93·9-s − 0.635·10-s − 0.745·11-s + 1.64·12-s − 5.62·13-s − 1.02·15-s + 1.74·16-s − 7.59·17-s + 1.23·18-s − 3.26·19-s − 1.59·20-s + 0.473·22-s + 6.27·23-s − 2.35·24-s + 25-s + 3.57·26-s + 5.08·27-s + 9.06·29-s + 0.654·30-s − 1.60·31-s − 5.67·32-s + ⋯ |
| L(s) = 1 | − 0.449·2-s − 0.594·3-s − 0.798·4-s + 0.447·5-s + 0.267·6-s + 0.807·8-s − 0.646·9-s − 0.200·10-s − 0.224·11-s + 0.474·12-s − 1.55·13-s − 0.265·15-s + 0.435·16-s − 1.84·17-s + 0.290·18-s − 0.749·19-s − 0.356·20-s + 0.101·22-s + 1.30·23-s − 0.480·24-s + 0.200·25-s + 0.700·26-s + 0.978·27-s + 1.68·29-s + 0.119·30-s − 0.287·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.635T + 2T^{2} \) |
| 3 | \( 1 + 1.02T + 3T^{2} \) |
| 11 | \( 1 + 0.745T + 11T^{2} \) |
| 13 | \( 1 + 5.62T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 - 9.06T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 0.303T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 4.17T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 - 4.44T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.29086200662192028666890427922, −6.79587276473309495858848077466, −5.99620001835102302564400105307, −5.15822917737091538971763632491, −4.78143346300154003642346165460, −4.13585292810074944327989229574, −2.76558294309614881458473947205, −2.26041550275904103940740726942, −0.856188545227526054592878222341, 0,
0.856188545227526054592878222341, 2.26041550275904103940740726942, 2.76558294309614881458473947205, 4.13585292810074944327989229574, 4.78143346300154003642346165460, 5.15822917737091538971763632491, 5.99620001835102302564400105307, 6.79587276473309495858848077466, 7.29086200662192028666890427922
