| L(s) = 1 | + 2-s − 2.93·3-s + 4-s + 3.18·5-s − 2.93·6-s + 3.42·7-s + 8-s + 5.61·9-s + 3.18·10-s − 4.68·11-s − 2.93·12-s − 2.68·13-s + 3.42·14-s − 9.36·15-s + 16-s − 1.49·17-s + 5.61·18-s − 1.25·19-s + 3.18·20-s − 10.0·21-s − 4.68·22-s + 5.10·23-s − 2.93·24-s + 5.17·25-s − 2.68·26-s − 7.68·27-s + 3.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.69·3-s + 0.5·4-s + 1.42·5-s − 1.19·6-s + 1.29·7-s + 0.353·8-s + 1.87·9-s + 1.00·10-s − 1.41·11-s − 0.847·12-s − 0.743·13-s + 0.915·14-s − 2.41·15-s + 0.250·16-s − 0.361·17-s + 1.32·18-s − 0.287·19-s + 0.713·20-s − 2.19·21-s − 0.998·22-s + 1.06·23-s − 0.599·24-s + 1.03·25-s − 0.525·26-s − 1.47·27-s + 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219156881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219156881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 4.68T + 11T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 - 5.47T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−13.13468634308794311107476945011, −12.60527978507983579335640904229, −11.13164288797226704626026517805, −10.88210562645405952793346246653, −9.666235384696241431687332688841, −7.62804042864662413549576591462, −6.35095734984476144792836712642, −5.18304759108051273320742286725, −5.02417646855981948872292047715, −1.99432093780152931135076713949,
1.99432093780152931135076713949, 5.02417646855981948872292047715, 5.18304759108051273320742286725, 6.35095734984476144792836712642, 7.62804042864662413549576591462, 9.666235384696241431687332688841, 10.88210562645405952793346246653, 11.13164288797226704626026517805, 12.60527978507983579335640904229, 13.13468634308794311107476945011
