| L(s) = 1 | + 1.65·2-s + 3.43·3-s + 0.739·4-s + 5-s + 5.68·6-s − 2.08·8-s + 8.77·9-s + 1.65·10-s − 11-s + 2.53·12-s + 4.15·13-s + 3.43·15-s − 4.93·16-s + 4.46·17-s + 14.5·18-s − 4.54·19-s + 0.739·20-s − 1.65·22-s − 6.25·23-s − 7.16·24-s + 25-s + 6.88·26-s + 19.8·27-s − 0.0247·29-s + 5.68·30-s − 0.999·31-s − 3.99·32-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 1.98·3-s + 0.369·4-s + 0.447·5-s + 2.31·6-s − 0.737·8-s + 2.92·9-s + 0.523·10-s − 0.301·11-s + 0.732·12-s + 1.15·13-s + 0.886·15-s − 1.23·16-s + 1.08·17-s + 3.42·18-s − 1.04·19-s + 0.165·20-s − 0.352·22-s − 1.30·23-s − 1.46·24-s + 0.200·25-s + 1.34·26-s + 3.81·27-s − 0.00459·29-s + 1.03·30-s − 0.179·31-s − 0.705·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.978825953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.978825953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.65T + 2T^{2} \) |
| 3 | \( 1 - 3.43T + 3T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 + 0.0247T + 29T^{2} \) |
| 31 | \( 1 + 0.999T + 31T^{2} \) |
| 37 | \( 1 + 0.802T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 - 0.771T + 43T^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.942T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 + 2.65T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.634634912172417430216452685923, −8.298617099630118994815254627873, −7.40272249005705247243151731347, −6.42794282100190093982196722551, −5.70953498189927848710031994847, −4.55029259536485401711688773295, −3.91227116747201617914144798914, −3.28169740664187136048205841883, −2.49360489457940496476849823295, −1.56784431073552010984795923440,
1.56784431073552010984795923440, 2.49360489457940496476849823295, 3.28169740664187136048205841883, 3.91227116747201617914144798914, 4.55029259536485401711688773295, 5.70953498189927848710031994847, 6.42794282100190093982196722551, 7.40272249005705247243151731347, 8.298617099630118994815254627873, 8.634634912172417430216452685923
