| L(s) = 1 | − 0.319·3-s − 2.89·9-s + 4.16·11-s − 2.89·13-s + 4.30·17-s − 1.95·19-s − 2.57·23-s + 1.88·27-s − 5.96·29-s − 9.43·31-s − 1.33·33-s − 6.32·37-s + 0.927·39-s + 7.19·41-s − 4.73·43-s + 4.43·47-s − 1.37·51-s + 7.35·53-s + 0.626·57-s + 3.22·59-s + 11.8·61-s + 11.6·67-s + 0.824·69-s + 7.58·71-s + 9.90·73-s − 12.8·79-s + 8.08·81-s + ⋯ |
| L(s) = 1 | − 0.184·3-s − 0.965·9-s + 1.25·11-s − 0.803·13-s + 1.04·17-s − 0.449·19-s − 0.537·23-s + 0.363·27-s − 1.10·29-s − 1.69·31-s − 0.232·33-s − 1.04·37-s + 0.148·39-s + 1.12·41-s − 0.722·43-s + 0.646·47-s − 0.192·51-s + 1.01·53-s + 0.0830·57-s + 0.419·59-s + 1.52·61-s + 1.42·67-s + 0.0992·69-s + 0.900·71-s + 1.15·73-s − 1.44·79-s + 0.898·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462831721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462831721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.319T + 3T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 9.43T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 - 9.90T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 - 3.01T + 89T^{2} \) |
| 97 | \( 1 + 0.414T + 97T^{2} \) |
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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
−7.59716004473845958961007538519, −6.99597875713489192556175032157, −6.30274637565635090643917983848, −5.48518945674019047659294116908, −5.21893932525817687924592330118, −3.90165943423040093042246214858, −3.67633518763919815680332640703, −2.52703873501999586784596716070, −1.76463652398962010646662689249, −0.57132320258953828564636696000,
0.57132320258953828564636696000, 1.76463652398962010646662689249, 2.52703873501999586784596716070, 3.67633518763919815680332640703, 3.90165943423040093042246214858, 5.21893932525817687924592330118, 5.48518945674019047659294116908, 6.30274637565635090643917983848, 6.99597875713489192556175032157, 7.59716004473845958961007538519
