L(s) = 1 | − 2.58·3-s + 5-s − 2.07·7-s + 3.68·9-s + 3.19·11-s − 5.95·13-s − 2.58·15-s − 6.46·17-s + 6.07·19-s + 5.37·21-s + 23-s + 25-s − 1.77·27-s − 3.87·29-s − 7.56·31-s − 8.26·33-s − 2.07·35-s − 2.64·37-s + 15.4·39-s − 0.979·41-s + 5.66·43-s + 3.68·45-s − 4.86·47-s − 2.68·49-s + 16.7·51-s + 8.53·53-s + 3.19·55-s + ⋯ |
L(s) = 1 | − 1.49·3-s + 0.447·5-s − 0.784·7-s + 1.22·9-s + 0.963·11-s − 1.65·13-s − 0.667·15-s − 1.56·17-s + 1.39·19-s + 1.17·21-s + 0.208·23-s + 0.200·25-s − 0.342·27-s − 0.720·29-s − 1.35·31-s − 1.43·33-s − 0.351·35-s − 0.434·37-s + 2.46·39-s − 0.153·41-s + 0.863·43-s + 0.549·45-s − 0.709·47-s − 0.383·49-s + 2.34·51-s + 1.17·53-s + 0.431·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5885950408\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5885950408\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 6.07T + 19T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 + 7.56T + 31T^{2} \) |
| 37 | \( 1 + 2.64T + 37T^{2} \) |
| 41 | \( 1 + 0.979T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 4.86T + 47T^{2} \) |
| 53 | \( 1 - 8.53T + 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 0.874T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−7.49588490442010978831792750212, −6.97471633316000245369766848784, −6.56048316256074195372247677798, −5.75413593857208739074389346744, −5.23127538271269174338941274500, −4.57245756571869968202242325793, −3.68803678498683518741029791363, −2.62250494277246858096456854380, −1.62995632702733711263361745804, −0.41492468125002068672631926197,
0.41492468125002068672631926197, 1.62995632702733711263361745804, 2.62250494277246858096456854380, 3.68803678498683518741029791363, 4.57245756571869968202242325793, 5.23127538271269174338941274500, 5.75413593857208739074389346744, 6.56048316256074195372247677798, 6.97471633316000245369766848784, 7.49588490442010978831792750212