| L(s) = 1 | + 2-s − 3-s + 4-s − 3.41·5-s − 6-s + 8-s + 9-s − 3.41·10-s − 3.82·11-s − 12-s − 1.82·13-s + 3.41·15-s + 16-s + 4.82·17-s + 18-s − 1.58·19-s − 3.41·20-s − 3.82·22-s + 6.65·23-s − 24-s + 6.65·25-s − 1.82·26-s − 27-s + 29-s + 3.41·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.07·10-s − 1.15·11-s − 0.288·12-s − 0.507·13-s + 0.881·15-s + 0.250·16-s + 1.17·17-s + 0.235·18-s − 0.363·19-s − 0.763·20-s − 0.816·22-s + 1.38·23-s − 0.204·24-s + 1.33·25-s − 0.358·26-s − 0.192·27-s + 0.185·29-s + 0.623·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + 0.585T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6.65T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.50545239223280786263231685800, −6.79365390349034685234019649079, −5.95647944618652095495897820727, −5.07414269236809496419717063867, −4.82242900176088647610258095792, −3.91585771839936739845473782954, −3.24275931804651064787365779158, −2.51307901176070494407394032130, −1.07503605648826395890640033352, 0,
1.07503605648826395890640033352, 2.51307901176070494407394032130, 3.24275931804651064787365779158, 3.91585771839936739845473782954, 4.82242900176088647610258095792, 5.07414269236809496419717063867, 5.95647944618652095495897820727, 6.79365390349034685234019649079, 7.50545239223280786263231685800
