| L(s) = 1 | + 2.37·2-s − 3-s + 3.65·4-s + 4.02·5-s − 2.37·6-s + 7-s + 3.92·8-s + 9-s + 9.57·10-s + 3.27·11-s − 3.65·12-s + 2.37·14-s − 4.02·15-s + 2.02·16-s − 0.377·17-s + 2.37·18-s + 3.54·19-s + 14.7·20-s − 21-s + 7.78·22-s + 1.89·23-s − 3.92·24-s + 11.2·25-s − 27-s + 3.65·28-s − 8.67·29-s − 9.57·30-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.82·4-s + 1.80·5-s − 0.970·6-s + 0.377·7-s + 1.38·8-s + 0.333·9-s + 3.02·10-s + 0.987·11-s − 1.05·12-s + 0.635·14-s − 1.04·15-s + 0.507·16-s − 0.0914·17-s + 0.560·18-s + 0.813·19-s + 3.28·20-s − 0.218·21-s + 1.65·22-s + 0.395·23-s − 0.801·24-s + 2.24·25-s − 0.192·27-s + 0.689·28-s − 1.61·29-s − 1.74·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.635811958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.635811958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 17 | \( 1 + 0.377T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 - 2.10T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 3.99T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 + 4.97T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.760704947612973201563687413559, −7.04802195279828171736019572459, −6.97027882700484109820538438746, −5.79151973835157365316797329305, −5.65816700240242021542346142750, −5.00260291304655767531995879396, −4.06432155851238166227655119491, −3.19352412319857796456294729873, −2.06448737829199994063437093168, −1.46949624989147591450027483292,
1.46949624989147591450027483292, 2.06448737829199994063437093168, 3.19352412319857796456294729873, 4.06432155851238166227655119491, 5.00260291304655767531995879396, 5.65816700240242021542346142750, 5.79151973835157365316797329305, 6.97027882700484109820538438746, 7.04802195279828171736019572459, 8.760704947612973201563687413559
