| L(s) = 1 | + 2.70·2-s + 0.0506·3-s + 5.32·4-s − 5-s + 0.137·6-s + 4.62·7-s + 9.01·8-s − 2.99·9-s − 2.70·10-s + 2.69·11-s + 0.270·12-s − 0.983·13-s + 12.5·14-s − 0.0506·15-s + 13.7·16-s − 4.02·17-s − 8.11·18-s − 5.32·20-s + 0.234·21-s + 7.30·22-s + 0.820·23-s + 0.457·24-s + 25-s − 2.66·26-s − 0.304·27-s + 24.6·28-s − 6.26·29-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.0292·3-s + 2.66·4-s − 0.447·5-s + 0.0560·6-s + 1.74·7-s + 3.18·8-s − 0.999·9-s − 0.856·10-s + 0.813·11-s + 0.0779·12-s − 0.272·13-s + 3.34·14-s − 0.0130·15-s + 3.43·16-s − 0.975·17-s − 1.91·18-s − 1.19·20-s + 0.0511·21-s + 1.55·22-s + 0.171·23-s + 0.0932·24-s + 0.200·25-s − 0.522·26-s − 0.0585·27-s + 4.65·28-s − 1.16·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.312573732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.312573732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.0506T + 3T^{2} \) |
| 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 0.983T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 23 | \( 1 - 0.820T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 + 2.21T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 - 2.06T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 8.99T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 0.461T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.091915141867195619296226498116, −8.197847195084494334837134060390, −7.48276502946539398426202948373, −6.67905212222829036183186935449, −5.73876532862114209380226387936, −5.06100270216883234917520115095, −4.38605558885247334833748405452, −3.65731755338429743870441140118, −2.51518233798438770010701030147, −1.64007936691672186734222479616,
1.64007936691672186734222479616, 2.51518233798438770010701030147, 3.65731755338429743870441140118, 4.38605558885247334833748405452, 5.06100270216883234917520115095, 5.73876532862114209380226387936, 6.67905212222829036183186935449, 7.48276502946539398426202948373, 8.197847195084494334837134060390, 9.091915141867195619296226498116
