L(s) = 1 | + 2.37·2-s + 3.66·4-s + 2.05·5-s + 5.08·7-s + 3.95·8-s + 4.90·10-s + 4.57·11-s − 1.87·13-s + 12.1·14-s + 2.08·16-s + 0.272·17-s + 0.546·19-s + 7.54·20-s + 10.8·22-s − 23-s − 0.759·25-s − 4.45·26-s + 18.6·28-s − 29-s − 4.25·31-s − 2.94·32-s + 0.647·34-s + 10.4·35-s + 1.52·37-s + 1.30·38-s + 8.14·40-s + 8.02·41-s + ⋯ |
L(s) = 1 | + 1.68·2-s + 1.83·4-s + 0.920·5-s + 1.92·7-s + 1.39·8-s + 1.54·10-s + 1.38·11-s − 0.518·13-s + 3.23·14-s + 0.521·16-s + 0.0660·17-s + 0.125·19-s + 1.68·20-s + 2.32·22-s − 0.208·23-s − 0.151·25-s − 0.873·26-s + 3.52·28-s − 0.185·29-s − 0.763·31-s − 0.520·32-s + 0.111·34-s + 1.77·35-s + 0.251·37-s + 0.210·38-s + 1.28·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.841475439\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.841475439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 - 0.272T + 17T^{2} \) |
| 19 | \( 1 - 0.546T + 19T^{2} \) |
| 31 | \( 1 + 4.25T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 9.38T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.81012035248640954563041027953, −7.18020751026867953366193079205, −6.34123746616249870040682512586, −5.74759458255423411533087152683, −5.13052589974969614232420609127, −4.50954131594816077368317419318, −3.95651089521564224219835083999, −2.89471274998213762181781908555, −1.85910698654649440966949804915, −1.54561915043543294176945714411,
1.54561915043543294176945714411, 1.85910698654649440966949804915, 2.89471274998213762181781908555, 3.95651089521564224219835083999, 4.50954131594816077368317419318, 5.13052589974969614232420609127, 5.74759458255423411533087152683, 6.34123746616249870040682512586, 7.18020751026867953366193079205, 7.81012035248640954563041027953