| L(s) = 1 | + 2.30·2-s + 3-s + 3.32·4-s + 2.75·5-s + 2.30·6-s − 1.67·7-s + 3.06·8-s + 9-s + 6.36·10-s + 4.71·11-s + 3.32·12-s − 2.29·13-s − 3.86·14-s + 2.75·15-s + 0.415·16-s − 17-s + 2.30·18-s − 1.88·19-s + 9.17·20-s − 1.67·21-s + 10.8·22-s + 0.872·23-s + 3.06·24-s + 2.60·25-s − 5.28·26-s + 27-s − 5.56·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.66·4-s + 1.23·5-s + 0.942·6-s − 0.632·7-s + 1.08·8-s + 0.333·9-s + 2.01·10-s + 1.42·11-s + 0.960·12-s − 0.635·13-s − 1.03·14-s + 0.712·15-s + 0.103·16-s − 0.242·17-s + 0.544·18-s − 0.431·19-s + 2.05·20-s − 0.365·21-s + 2.32·22-s + 0.181·23-s + 0.625·24-s + 0.521·25-s − 1.03·26-s + 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.692012848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.692012848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 23 | \( 1 - 0.872T + 23T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 - 0.386T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 4.93T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 - 0.942T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 7.61T + 97T^{2} \) |
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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
−8.538963051683628844575155750098, −7.31742115401154399293136260104, −6.54469143879682064473500216733, −6.24880177923825764013392406657, −5.48115873105972838695194257157, −4.46238168453291007549328116263, −4.04400554059368425252201233543, −2.88711398842737040075906552039, −2.49387981305551149614471915411, −1.39103385606752393935469119015,
1.39103385606752393935469119015, 2.49387981305551149614471915411, 2.88711398842737040075906552039, 4.04400554059368425252201233543, 4.46238168453291007549328116263, 5.48115873105972838695194257157, 6.24880177923825764013392406657, 6.54469143879682064473500216733, 7.31742115401154399293136260104, 8.538963051683628844575155750098
