| L(s) = 1 | + 2-s + 1.15·3-s + 4-s + 2.73·5-s + 1.15·6-s − 7-s + 8-s − 1.67·9-s + 2.73·10-s + 0.805·11-s + 1.15·12-s + 3.64·13-s − 14-s + 3.15·15-s + 16-s + 6.42·17-s − 1.67·18-s + 0.0107·19-s + 2.73·20-s − 1.15·21-s + 0.805·22-s + 1.19·23-s + 1.15·24-s + 2.48·25-s + 3.64·26-s − 5.38·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.664·3-s + 0.5·4-s + 1.22·5-s + 0.470·6-s − 0.377·7-s + 0.353·8-s − 0.557·9-s + 0.865·10-s + 0.242·11-s + 0.332·12-s + 1.01·13-s − 0.267·14-s + 0.813·15-s + 0.250·16-s + 1.55·17-s − 0.394·18-s + 0.00247·19-s + 0.611·20-s − 0.251·21-s + 0.171·22-s + 0.249·23-s + 0.235·24-s + 0.497·25-s + 0.714·26-s − 1.03·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.435554227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.435554227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 0.805T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 - 0.0107T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 - 0.284T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.35T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 2.33T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
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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
−8.164120882611029129425482145981, −7.28624367890043818860957419546, −6.34880016375761647469295975577, −5.96173535464395070125587163108, −5.37230857062044304550702269577, −4.42057998460534169737669009868, −3.28554539232589446578583325312, −3.08335072592155667620801242181, −2.02197239526414506988504892814, −1.15827914185105050478494293030,
1.15827914185105050478494293030, 2.02197239526414506988504892814, 3.08335072592155667620801242181, 3.28554539232589446578583325312, 4.42057998460534169737669009868, 5.37230857062044304550702269577, 5.96173535464395070125587163108, 6.34880016375761647469295975577, 7.28624367890043818860957419546, 8.164120882611029129425482145981
