| L(s) = 1 | − 1.21·2-s + 2.33·3-s − 0.512·4-s + 5-s − 2.84·6-s − 3.60·7-s + 3.06·8-s + 2.43·9-s − 1.21·10-s − 5.37·11-s − 1.19·12-s + 4.39·14-s + 2.33·15-s − 2.71·16-s − 1.13·17-s − 2.97·18-s − 2.26·19-s − 0.512·20-s − 8.39·21-s + 6.55·22-s − 3.89·23-s + 7.14·24-s + 25-s − 1.30·27-s + 1.84·28-s − 0.0247·29-s − 2.84·30-s + ⋯ |
| L(s) = 1 | − 0.862·2-s + 1.34·3-s − 0.256·4-s + 0.447·5-s − 1.16·6-s − 1.36·7-s + 1.08·8-s + 0.813·9-s − 0.385·10-s − 1.61·11-s − 0.344·12-s + 1.17·14-s + 0.602·15-s − 0.678·16-s − 0.274·17-s − 0.701·18-s − 0.520·19-s − 0.114·20-s − 1.83·21-s + 1.39·22-s − 0.811·23-s + 1.45·24-s + 0.200·25-s − 0.251·27-s + 0.348·28-s − 0.00459·29-s − 0.519·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 + 0.0247T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 - 0.171T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 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
−9.625829059469435198820628255089, −9.042693443659476638159445516824, −8.167800203022292438696664529422, −7.67340167559584802764663393421, −6.55254510447461649334817236883, −5.35761659014663467352607282305, −4.04299855008197229101909617313, −2.96804483755670435285527783189, −2.08415832884535496228005583226, 0,
2.08415832884535496228005583226, 2.96804483755670435285527783189, 4.04299855008197229101909617313, 5.35761659014663467352607282305, 6.55254510447461649334817236883, 7.67340167559584802764663393421, 8.167800203022292438696664529422, 9.042693443659476638159445516824, 9.625829059469435198820628255089
