| L(s) = 1 | + 4.29·5-s + 0.828·7-s − 3.56·11-s + 2·13-s + 0.737·17-s + 2.82·19-s + 4.29·23-s + 13.4·25-s + 0.737·29-s − 6.82·31-s + 3.56·35-s − 37-s − 5.03·41-s + 2.82·43-s + 5.03·47-s − 6.31·49-s + 3.56·53-s − 15.3·55-s + 7.86·59-s + 7.65·61-s + 8.59·65-s − 8.82·67-s + 12.1·71-s + 14.8·73-s − 2.95·77-s + 8.48·79-s − 15.7·83-s + ⋯ |
| L(s) = 1 | + 1.92·5-s + 0.313·7-s − 1.07·11-s + 0.554·13-s + 0.178·17-s + 0.648·19-s + 0.896·23-s + 2.69·25-s + 0.136·29-s − 1.22·31-s + 0.602·35-s − 0.164·37-s − 0.786·41-s + 0.431·43-s + 0.734·47-s − 0.901·49-s + 0.489·53-s − 2.06·55-s + 1.02·59-s + 0.980·61-s + 1.06·65-s − 1.07·67-s + 1.44·71-s + 1.73·73-s − 0.336·77-s + 0.954·79-s − 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.165784528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.165784528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 0.737T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 0.737T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.345598349646763143862589098945, −7.34715972407034677233597625712, −6.72228971296591235987199889273, −5.84100956015699275755683512642, −5.36297544752853678557256427609, −4.88969946390402588184835878196, −3.52933146234904830267590787887, −2.64247199657249976136644436465, −1.93787951646035408450995947282, −1.01314474752714126493460891235,
1.01314474752714126493460891235, 1.93787951646035408450995947282, 2.64247199657249976136644436465, 3.52933146234904830267590787887, 4.88969946390402588184835878196, 5.36297544752853678557256427609, 5.84100956015699275755683512642, 6.72228971296591235987199889273, 7.34715972407034677233597625712, 8.345598349646763143862589098945
