| L(s) = 1 | + 3-s + 2.40·5-s + 3.93·7-s + 9-s − 5.83·11-s + 4.21·13-s + 2.40·15-s + 0.537·17-s + 5.08·19-s + 3.93·21-s − 23-s + 0.798·25-s + 27-s + 29-s + 1.84·31-s − 5.83·33-s + 9.46·35-s − 6.96·37-s + 4.21·39-s − 0.514·41-s + 10.1·43-s + 2.40·45-s + 4.15·47-s + 8.45·49-s + 0.537·51-s − 10.8·53-s − 14.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.07·5-s + 1.48·7-s + 0.333·9-s − 1.76·11-s + 1.16·13-s + 0.621·15-s + 0.130·17-s + 1.16·19-s + 0.857·21-s − 0.208·23-s + 0.159·25-s + 0.192·27-s + 0.185·29-s + 0.332·31-s − 1.01·33-s + 1.60·35-s − 1.14·37-s + 0.675·39-s − 0.0804·41-s + 1.54·43-s + 0.358·45-s + 0.606·47-s + 1.20·49-s + 0.0752·51-s − 1.49·53-s − 1.89·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.076779777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.076779777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 5.83T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 - 0.537T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 + 0.514T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 + 6.51T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 + 5.46T + 97T^{2} \) |
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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
−7.79576849051757270217068349358, −7.47942073207461703721052288264, −6.36676202924918943888879255581, −5.47240838178729118887811045472, −5.28003574898153748101478236962, −4.38087342694648232974253270635, −3.37162142638902399624557883677, −2.50935539659109401829306813590, −1.86881430119024689316058755898, −1.04267117757677241481505014007,
1.04267117757677241481505014007, 1.86881430119024689316058755898, 2.50935539659109401829306813590, 3.37162142638902399624557883677, 4.38087342694648232974253270635, 5.28003574898153748101478236962, 5.47240838178729118887811045472, 6.36676202924918943888879255581, 7.47942073207461703721052288264, 7.79576849051757270217068349358
