| L(s) = 1 | + 2.21·3-s − 5-s + 2.83·7-s + 1.90·9-s − 1.37·11-s + 4.28·13-s − 2.21·15-s − 3.33·17-s + 8.21·19-s + 6.28·21-s − 0.622·23-s + 25-s − 2.42·27-s − 5.18·29-s + 6.64·31-s − 3.05·33-s − 2.83·35-s − 37-s + 9.47·39-s + 2.42·41-s − 1.18·43-s − 1.90·45-s + 2.54·47-s + 1.04·49-s − 7.37·51-s − 2.56·53-s + 1.37·55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s − 0.447·5-s + 1.07·7-s + 0.634·9-s − 0.415·11-s + 1.18·13-s − 0.571·15-s − 0.808·17-s + 1.88·19-s + 1.37·21-s − 0.129·23-s + 0.200·25-s − 0.467·27-s − 0.962·29-s + 1.19·31-s − 0.531·33-s − 0.479·35-s − 0.164·37-s + 1.51·39-s + 0.379·41-s − 0.180·43-s − 0.283·45-s + 0.370·47-s + 0.149·49-s − 1.03·51-s − 0.351·53-s + 0.185·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.424402340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424402340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 7 | \( 1 - 2.83T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 + 0.622T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 8.87T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.622T + 73T^{2} \) |
| 79 | \( 1 + 9.26T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−10.33013770599285273382943093376, −9.219766453059109621326389419320, −8.606031445157603895972500082029, −7.88732067957471548527583511297, −7.32946555558656681313014232762, −5.88067081928386728704812378843, −4.74106964991201324251356936378, −3.72272476149387066681604674080, −2.78548965009449560310630848329, −1.47808082823774441614281911443,
1.47808082823774441614281911443, 2.78548965009449560310630848329, 3.72272476149387066681604674080, 4.74106964991201324251356936378, 5.88067081928386728704812378843, 7.32946555558656681313014232762, 7.88732067957471548527583511297, 8.606031445157603895972500082029, 9.219766453059109621326389419320, 10.33013770599285273382943093376
