| L(s) = 1 | − 1.65·3-s + 0.480·5-s − 0.613·7-s − 0.271·9-s + 1.38·11-s − 2.98·13-s − 0.794·15-s + 8.14·17-s + 19-s + 1.01·21-s + 0.301·23-s − 4.76·25-s + 5.40·27-s − 1.51·29-s − 2.09·31-s − 2.28·33-s − 0.295·35-s − 5.16·37-s + 4.93·39-s + 2.24·41-s − 4.94·43-s − 0.130·45-s − 1.78·47-s − 6.62·49-s − 13.4·51-s + 53-s + 0.665·55-s + ⋯ |
| L(s) = 1 | − 0.953·3-s + 0.215·5-s − 0.231·7-s − 0.0904·9-s + 0.417·11-s − 0.827·13-s − 0.205·15-s + 1.97·17-s + 0.229·19-s + 0.221·21-s + 0.0627·23-s − 0.953·25-s + 1.03·27-s − 0.280·29-s − 0.375·31-s − 0.398·33-s − 0.0498·35-s − 0.849·37-s + 0.789·39-s + 0.351·41-s − 0.754·43-s − 0.0194·45-s − 0.260·47-s − 0.946·49-s − 1.88·51-s + 0.137·53-s + 0.0897·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110640440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110640440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 - 0.480T + 5T^{2} \) |
| 7 | \( 1 + 0.613T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 - 8.14T + 17T^{2} \) |
| 23 | \( 1 - 0.301T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 - 3.10T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 0.261T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.41T + 97T^{2} \) |
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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
−8.362961623665187434289097654654, −7.62317194082492791785165123445, −6.89202159033485470510708146070, −6.11487741746378912742961744924, −5.44438848481266857526032970075, −5.01968052701514830951160373709, −3.80935226603403775266481332192, −3.05160710091137838900566530953, −1.82034095069597439199013748317, −0.63314068680705269477803512659,
0.63314068680705269477803512659, 1.82034095069597439199013748317, 3.05160710091137838900566530953, 3.80935226603403775266481332192, 5.01968052701514830951160373709, 5.44438848481266857526032970075, 6.11487741746378912742961744924, 6.89202159033485470510708146070, 7.62317194082492791785165123445, 8.362961623665187434289097654654
