| L(s) = 1 | + 3-s − 0.488·5-s + 0.956·7-s + 9-s − 2.60·11-s − 2.95·13-s − 0.488·15-s + 1.58·17-s + 3.22·19-s + 0.956·21-s − 4.40·23-s − 4.76·25-s + 27-s + 1.02·29-s − 6.26·31-s − 2.60·33-s − 0.467·35-s − 6.38·37-s − 2.95·39-s − 41-s + 0.820·43-s − 0.488·45-s − 3.15·47-s − 6.08·49-s + 1.58·51-s + 3.62·53-s + 1.27·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.218·5-s + 0.361·7-s + 0.333·9-s − 0.784·11-s − 0.819·13-s − 0.126·15-s + 0.383·17-s + 0.740·19-s + 0.208·21-s − 0.917·23-s − 0.952·25-s + 0.192·27-s + 0.190·29-s − 1.12·31-s − 0.453·33-s − 0.0789·35-s − 1.04·37-s − 0.473·39-s − 0.156·41-s + 0.125·43-s − 0.0728·45-s − 0.460·47-s − 0.869·49-s + 0.221·51-s + 0.498·53-s + 0.171·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 0.488T + 5T^{2} \) |
| 7 | \( 1 - 0.956T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 43 | \( 1 - 0.820T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.40T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−7.988962212275682877585913452802, −7.55694054161626820699899523849, −6.86751089160481415153085499698, −5.67175772037678886613253850707, −5.15086717565242422855020360022, −4.18937689479734185993145366329, −3.39467776409956153963177889514, −2.48625421452640073404724289358, −1.61443133291253230807684820128, 0,
1.61443133291253230807684820128, 2.48625421452640073404724289358, 3.39467776409956153963177889514, 4.18937689479734185993145366329, 5.15086717565242422855020360022, 5.67175772037678886613253850707, 6.86751089160481415153085499698, 7.55694054161626820699899523849, 7.988962212275682877585913452802
