| L(s) = 1 | − 3-s − 2.61·5-s − 3·7-s + 9-s + 1.76·13-s + 2.61·15-s + 1.61·17-s − 5.85·19-s + 3·21-s − 3.47·23-s + 1.85·25-s − 27-s + 4.47·29-s − 2.85·31-s + 7.85·35-s + 0.236·37-s − 1.76·39-s − 11.9·41-s − 6.23·43-s − 2.61·45-s − 1.61·47-s + 2·49-s − 1.61·51-s − 9.61·53-s + 5.85·57-s − 10.3·59-s + 7.85·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.17·5-s − 1.13·7-s + 0.333·9-s + 0.489·13-s + 0.675·15-s + 0.392·17-s − 1.34·19-s + 0.654·21-s − 0.723·23-s + 0.370·25-s − 0.192·27-s + 0.830·29-s − 0.512·31-s + 1.32·35-s + 0.0388·37-s − 0.282·39-s − 1.86·41-s − 0.950·43-s − 0.390·45-s − 0.236·47-s + 0.285·49-s − 0.226·51-s − 1.32·53-s + 0.775·57-s − 1.34·59-s + 1.00·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3810078306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3810078306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 - 0.527T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.227677073301440636778587645486, −7.30145075560709680016258265956, −6.56252347741541329568994263668, −6.21814035549411703469023263445, −5.20806012214441834281969011902, −4.35266707509660258513395436549, −3.69104202450473518827236482306, −3.07775665047410773766177408364, −1.74000988766853428659862717657, −0.32801788998822681180893444069,
0.32801788998822681180893444069, 1.74000988766853428659862717657, 3.07775665047410773766177408364, 3.69104202450473518827236482306, 4.35266707509660258513395436549, 5.20806012214441834281969011902, 6.21814035549411703469023263445, 6.56252347741541329568994263668, 7.30145075560709680016258265956, 8.227677073301440636778587645486
