| L(s) = 1 | + 2-s + 0.969·3-s + 4-s − 3.11·5-s + 0.969·6-s + 7-s + 8-s − 2.06·9-s − 3.11·10-s + 0.0462·11-s + 0.969·12-s + 7.03·13-s + 14-s − 3.01·15-s + 16-s + 4.97·17-s − 2.06·18-s − 3.11·20-s + 0.969·21-s + 0.0462·22-s − 1.89·23-s + 0.969·24-s + 4.67·25-s + 7.03·26-s − 4.90·27-s + 28-s − 4.28·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.559·3-s + 0.5·4-s − 1.39·5-s + 0.395·6-s + 0.377·7-s + 0.353·8-s − 0.686·9-s − 0.983·10-s + 0.0139·11-s + 0.279·12-s + 1.95·13-s + 0.267·14-s − 0.778·15-s + 0.250·16-s + 1.20·17-s − 0.485·18-s − 0.695·20-s + 0.211·21-s + 0.00986·22-s − 0.394·23-s + 0.197·24-s + 0.935·25-s + 1.37·26-s − 0.943·27-s + 0.188·28-s − 0.796·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.164208496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.164208496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.969T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 11 | \( 1 - 0.0462T + 11T^{2} \) |
| 13 | \( 1 - 7.03T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 + 6.57T + 37T^{2} \) |
| 41 | \( 1 - 8.25T + 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 - 0.348T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−8.039429847485717972090380465106, −7.75042717571698397870958197268, −6.85661831453922599731786142702, −5.84690125559354049576348163019, −5.41877402066887097197372948864, −4.20260381453364587823394091388, −3.65410911077577153032892544006, −3.30461199069004736091489204999, −2.09347462593206166201481577025, −0.864949220452367541968341770690,
0.864949220452367541968341770690, 2.09347462593206166201481577025, 3.30461199069004736091489204999, 3.65410911077577153032892544006, 4.20260381453364587823394091388, 5.41877402066887097197372948864, 5.84690125559354049576348163019, 6.85661831453922599731786142702, 7.75042717571698397870958197268, 8.039429847485717972090380465106
