| L(s) = 1 | + 1.58·2-s + 9.82·3-s − 5.48·4-s − 16.5·5-s + 15.5·6-s − 21.3·8-s + 69.5·9-s − 26.2·10-s + 19.9·11-s − 53.8·12-s − 6.07·13-s − 162.·15-s + 9.89·16-s + 95.1·17-s + 110.·18-s − 82.1·19-s + 90.4·20-s + 31.6·22-s − 159.·23-s − 210.·24-s + 147.·25-s − 9.63·26-s + 418.·27-s + 120.·29-s − 257.·30-s − 109.·31-s + 186.·32-s + ⋯ |
| L(s) = 1 | + 0.561·2-s + 1.89·3-s − 0.685·4-s − 1.47·5-s + 1.06·6-s − 0.945·8-s + 2.57·9-s − 0.828·10-s + 0.547·11-s − 1.29·12-s − 0.129·13-s − 2.79·15-s + 0.154·16-s + 1.35·17-s + 1.44·18-s − 0.992·19-s + 1.01·20-s + 0.307·22-s − 1.44·23-s − 1.78·24-s + 1.18·25-s − 0.0727·26-s + 2.98·27-s + 0.769·29-s − 1.56·30-s − 0.635·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.699707098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.699707098\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 1.58T + 8T^{2} \) |
| 3 | \( 1 - 9.82T + 27T^{2} \) |
| 5 | \( 1 + 16.5T + 125T^{2} \) |
| 11 | \( 1 - 19.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.07T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 254.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 103.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 7.53T + 1.48e5T^{2} \) |
| 59 | \( 1 - 699.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 978.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 940.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 356.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 801.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 969.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 97.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 967.T + 9.12e5T^{2} \) |
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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
−8.386864294965106485349598073997, −8.150938021703517581007324988377, −7.44302743981032616068596371916, −6.50600130034211832772559796298, −5.13227954454788551073398990695, −4.09415218021488339239313736258, −3.84355919494546181664620807541, −3.24564158044095741631374104645, −2.12939247378757148462856260237, −0.72155823900771293969967243686,
0.72155823900771293969967243686, 2.12939247378757148462856260237, 3.24564158044095741631374104645, 3.84355919494546181664620807541, 4.09415218021488339239313736258, 5.13227954454788551073398990695, 6.50600130034211832772559796298, 7.44302743981032616068596371916, 8.150938021703517581007324988377, 8.386864294965106485349598073997
