| L(s) = 1 | + 2.79·3-s − 5-s + 1.79·7-s + 4.79·9-s + 0.791·11-s + 5.79·13-s − 2.79·15-s + 0.791·17-s − 5.79·19-s + 5·21-s − 23-s + 25-s + 4.99·27-s + 7.58·29-s + 3.37·31-s + 2.20·33-s − 1.79·35-s − 4·37-s + 16.1·39-s − 6.79·41-s − 11.1·43-s − 4.79·45-s + 4.41·47-s − 3.79·49-s + 2.20·51-s + 6·53-s − 0.791·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 0.447·5-s + 0.677·7-s + 1.59·9-s + 0.238·11-s + 1.60·13-s − 0.720·15-s + 0.191·17-s − 1.32·19-s + 1.09·21-s − 0.208·23-s + 0.200·25-s + 0.962·27-s + 1.40·29-s + 0.605·31-s + 0.384·33-s − 0.302·35-s − 0.657·37-s + 2.58·39-s − 1.06·41-s − 1.70·43-s − 0.714·45-s + 0.644·47-s − 0.541·49-s + 0.309·51-s + 0.824·53-s − 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.331190207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.331190207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 - 0.791T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 7.95T + 97T^{2} \) |
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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
−8.744147764448629160552381936385, −8.515253642608303670110376157663, −8.105654653243987316633960702933, −7.02393613300488824488128644459, −6.28370926587918452379634881985, −4.90763234032842049286194293711, −3.97925541850641924941011032608, −3.43266895891183257743930666625, −2.31734844278758075461923703741, −1.32419911688698329532723629697,
1.32419911688698329532723629697, 2.31734844278758075461923703741, 3.43266895891183257743930666625, 3.97925541850641924941011032608, 4.90763234032842049286194293711, 6.28370926587918452379634881985, 7.02393613300488824488128644459, 8.105654653243987316633960702933, 8.515253642608303670110376157663, 8.744147764448629160552381936385
