| L(s) = 1 | + 3-s + 1.61·5-s + 7-s + 9-s + 2.23·11-s − 4.61·13-s + 1.61·15-s − 6.70·17-s − 5.47·19-s + 21-s + 23-s − 2.38·25-s + 27-s − 3.76·29-s + 6.70·31-s + 2.23·33-s + 1.61·35-s − 11·37-s − 4.61·39-s + 7.47·41-s − 0.618·43-s + 1.61·45-s + 2.76·47-s + 49-s − 6.70·51-s − 1.90·53-s + 3.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.723·5-s + 0.377·7-s + 0.333·9-s + 0.674·11-s − 1.28·13-s + 0.417·15-s − 1.62·17-s − 1.25·19-s + 0.218·21-s + 0.208·23-s − 0.476·25-s + 0.192·27-s − 0.698·29-s + 1.20·31-s + 0.389·33-s + 0.273·35-s − 1.80·37-s − 0.739·39-s + 1.16·41-s − 0.0942·43-s + 0.241·45-s + 0.403·47-s + 0.142·49-s − 0.939·51-s − 0.262·53-s + 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 + 0.618T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.49174185690278150122121328506, −6.81108386078542963819171176447, −6.27717675271568572783353313241, −5.38709796497648731496636267061, −4.49096009984607168165422525718, −4.14736443020660706056601097468, −2.88978472574462379918799319639, −2.18739715156204261957333036861, −1.62282548934540466802950009127, 0,
1.62282548934540466802950009127, 2.18739715156204261957333036861, 2.88978472574462379918799319639, 4.14736443020660706056601097468, 4.49096009984607168165422525718, 5.38709796497648731496636267061, 6.27717675271568572783353313241, 6.81108386078542963819171176447, 7.49174185690278150122121328506
