| L(s) = 1 | + 0.104·2-s + 0.696·3-s − 1.98·4-s + 0.0725·6-s − 3.21·7-s − 0.415·8-s − 2.51·9-s − 11-s − 1.38·12-s − 4.65·13-s − 0.334·14-s + 3.93·16-s + 0.00602·17-s − 0.261·18-s − 19-s − 2.24·21-s − 0.104·22-s + 1.65·23-s − 0.289·24-s − 0.484·26-s − 3.84·27-s + 6.39·28-s + 3.49·29-s − 3.34·31-s + 1.24·32-s − 0.696·33-s + 0.000627·34-s + ⋯ |
| L(s) = 1 | + 0.0736·2-s + 0.402·3-s − 0.994·4-s + 0.0296·6-s − 1.21·7-s − 0.146·8-s − 0.838·9-s − 0.301·11-s − 0.400·12-s − 1.29·13-s − 0.0895·14-s + 0.983·16-s + 0.00146·17-s − 0.0617·18-s − 0.229·19-s − 0.489·21-s − 0.0222·22-s + 0.345·23-s − 0.0591·24-s − 0.0950·26-s − 0.739·27-s + 1.20·28-s + 0.649·29-s − 0.600·31-s + 0.219·32-s − 0.121·33-s + 0.000107·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4588255891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4588255891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.104T + 2T^{2} \) |
| 3 | \( 1 - 0.696T + 3T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 - 0.00602T + 17T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 + 0.893T + 47T^{2} \) |
| 53 | \( 1 + 1.78T + 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 - 0.852T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.394523024923052674617185766375, −7.55631758301602478388909700722, −6.78916747437646939623409021612, −5.98188763320366041690448127255, −5.17616087120504242662362980743, −4.63815978632555991581915770045, −3.36694718064520745619864332223, −3.20714069065155209090285381049, −2.06071780952250466796985922724, −0.33432733389014779274322184622,
0.33432733389014779274322184622, 2.06071780952250466796985922724, 3.20714069065155209090285381049, 3.36694718064520745619864332223, 4.63815978632555991581915770045, 5.17616087120504242662362980743, 5.98188763320366041690448127255, 6.78916747437646939623409021612, 7.55631758301602478388909700722, 8.394523024923052674617185766375
