| L(s) = 1 | + 2.71·2-s + 3-s + 5.38·4-s + 2.08·5-s + 2.71·6-s + 0.469·7-s + 9.18·8-s + 9-s + 5.67·10-s + 2.70·11-s + 5.38·12-s − 6.47·13-s + 1.27·14-s + 2.08·15-s + 14.1·16-s − 1.65·17-s + 2.71·18-s − 6.08·19-s + 11.2·20-s + 0.469·21-s + 7.35·22-s − 0.669·23-s + 9.18·24-s − 0.642·25-s − 17.5·26-s + 27-s + 2.52·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 0.577·3-s + 2.69·4-s + 0.933·5-s + 1.10·6-s + 0.177·7-s + 3.24·8-s + 0.333·9-s + 1.79·10-s + 0.816·11-s + 1.55·12-s − 1.79·13-s + 0.340·14-s + 0.538·15-s + 3.54·16-s − 0.401·17-s + 0.640·18-s − 1.39·19-s + 2.51·20-s + 0.102·21-s + 1.56·22-s − 0.139·23-s + 1.87·24-s − 0.128·25-s − 3.44·26-s + 0.192·27-s + 0.477·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.016133468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.016133468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 0.469T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 + 0.669T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 - 1.97T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 - 4.51T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 8.64T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 0.182T + 73T^{2} \) |
| 79 | \( 1 - 0.130T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−8.718962056185878313460401126358, −7.68457473308317917934813165045, −6.86973864368858916121320506395, −6.41904446437774815667580945836, −5.53737797955511942120850013280, −4.74093650664503659438361308806, −4.21099081439677581746050618286, −3.18204180185510272372790443762, −2.25600601611750982820997743030, −1.84503252448462112718915261599,
1.84503252448462112718915261599, 2.25600601611750982820997743030, 3.18204180185510272372790443762, 4.21099081439677581746050618286, 4.74093650664503659438361308806, 5.53737797955511942120850013280, 6.41904446437774815667580945836, 6.86973864368858916121320506395, 7.68457473308317917934813165045, 8.718962056185878313460401126358
