| L(s) = 1 | + 2.38·2-s + 3.09·3-s + 3.69·4-s + 5-s + 7.38·6-s + 4.05·8-s + 6.56·9-s + 2.38·10-s + 0.408·11-s + 11.4·12-s − 4.28·13-s + 3.09·15-s + 2.27·16-s − 5.26·17-s + 15.6·18-s + 6.34·19-s + 3.69·20-s + 0.975·22-s + 9.14·23-s + 12.5·24-s + 25-s − 10.2·26-s + 11.0·27-s + 29-s + 7.38·30-s − 7.04·31-s − 2.67·32-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.78·3-s + 1.84·4-s + 0.447·5-s + 3.01·6-s + 1.43·8-s + 2.18·9-s + 0.754·10-s + 0.123·11-s + 3.29·12-s − 1.18·13-s + 0.798·15-s + 0.568·16-s − 1.27·17-s + 3.69·18-s + 1.45·19-s + 0.826·20-s + 0.207·22-s + 1.90·23-s + 2.55·24-s + 0.200·25-s − 2.00·26-s + 2.11·27-s + 0.185·29-s + 1.34·30-s − 1.26·31-s − 0.472·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.09858836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.09858836\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 11 | \( 1 - 0.408T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 5.26T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 9.14T + 23T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 - 0.875T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−7.53087351265623940319843226943, −7.24499422907129619137773254667, −6.64800526807158264690749203653, −5.54426879712651717997098053915, −4.90344401390659145357253964895, −4.29078408162505419315771013117, −3.48884732698479571699048220369, −2.78048305281878839098022820550, −2.44778463495948121605077248044, −1.47292203168536355048998479226,
1.47292203168536355048998479226, 2.44778463495948121605077248044, 2.78048305281878839098022820550, 3.48884732698479571699048220369, 4.29078408162505419315771013117, 4.90344401390659145357253964895, 5.54426879712651717997098053915, 6.64800526807158264690749203653, 7.24499422907129619137773254667, 7.53087351265623940319843226943
