| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s + 2.82·13-s − 15-s − 6.82·17-s + 2.58·19-s − 4.58·23-s + 25-s + 27-s − 7.65·29-s − 4.24·31-s − 2·33-s + 6.48·37-s + 2.82·39-s − 2·41-s − 9.65·43-s − 45-s + 6.48·47-s − 6.82·51-s − 7.41·53-s + 2·55-s + 2.58·57-s − 2.82·59-s + 9.89·61-s − 2.82·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.333·9-s − 0.603·11-s + 0.784·13-s − 0.258·15-s − 1.65·17-s + 0.593·19-s − 0.956·23-s + 0.200·25-s + 0.192·27-s − 1.42·29-s − 0.762·31-s − 0.348·33-s + 1.06·37-s + 0.452·39-s − 0.312·41-s − 1.47·43-s − 0.149·45-s + 0.945·47-s − 0.956·51-s − 1.01·53-s + 0.269·55-s + 0.342·57-s − 0.368·59-s + 1.26·61-s − 0.350·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 7.41T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 0.828T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.361335215979670970569475964319, −7.72673085161866565159171812303, −7.00283509439708003039241775247, −6.14794912890750734565503182837, −5.23919646560027316425032398702, −4.24756305440851036397164499021, −3.63618284111658868687469611819, −2.61333741625950223822442628484, −1.66115422113308758966633808171, 0,
1.66115422113308758966633808171, 2.61333741625950223822442628484, 3.63618284111658868687469611819, 4.24756305440851036397164499021, 5.23919646560027316425032398702, 6.14794912890750734565503182837, 7.00283509439708003039241775247, 7.72673085161866565159171812303, 8.361335215979670970569475964319
