| L(s) = 1 | + 2-s − 0.356·3-s + 4-s + 3.98·5-s − 0.356·6-s + 1.62·7-s + 8-s − 2.87·9-s + 3.98·10-s + 2.63·11-s − 0.356·12-s + 6.02·13-s + 1.62·14-s − 1.42·15-s + 16-s − 0.903·17-s − 2.87·18-s + 4.65·19-s + 3.98·20-s − 0.580·21-s + 2.63·22-s + 23-s − 0.356·24-s + 10.8·25-s + 6.02·26-s + 2.09·27-s + 1.62·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.205·3-s + 0.5·4-s + 1.78·5-s − 0.145·6-s + 0.615·7-s + 0.353·8-s − 0.957·9-s + 1.26·10-s + 0.795·11-s − 0.102·12-s + 1.67·13-s + 0.435·14-s − 0.366·15-s + 0.250·16-s − 0.219·17-s − 0.677·18-s + 1.06·19-s + 0.891·20-s − 0.126·21-s + 0.562·22-s + 0.208·23-s − 0.0726·24-s + 2.17·25-s + 1.18·26-s + 0.402·27-s + 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.164732821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.164732821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.356T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 + 0.903T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 29 | \( 1 - 0.600T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 + 8.50T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 4.11T + 43T^{2} \) |
| 47 | \( 1 + 2.46T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 - 0.448T + 59T^{2} \) |
| 61 | \( 1 + 0.851T + 61T^{2} \) |
| 67 | \( 1 + 7.36T + 67T^{2} \) |
| 71 | \( 1 + 2.42T + 71T^{2} \) |
| 73 | \( 1 - 7.80T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.204387763828114414923831800728, −7.01352736710541187693924366304, −6.39924188978808601253892859781, −5.81765221018706041329447902412, −5.42131154588015469434673203830, −4.65875925146921397683675213020, −3.53855682546426799928053534503, −2.86924504330212465699757546164, −1.74925567346198717505516984448, −1.27127359008317085929097633338,
1.27127359008317085929097633338, 1.74925567346198717505516984448, 2.86924504330212465699757546164, 3.53855682546426799928053534503, 4.65875925146921397683675213020, 5.42131154588015469434673203830, 5.81765221018706041329447902412, 6.39924188978808601253892859781, 7.01352736710541187693924366304, 8.204387763828114414923831800728
