| L(s) = 1 | − 2.24·3-s + 3.59·5-s + 1.86·7-s + 2.04·9-s − 1.16·11-s + 1.95·13-s − 8.06·15-s + 4.16·17-s + 6.65·19-s − 4.19·21-s − 6.60·23-s + 7.90·25-s + 2.14·27-s + 4.88·29-s + 9.12·31-s + 2.60·33-s + 6.70·35-s + 4.73·37-s − 4.38·39-s − 0.418·41-s − 10.0·43-s + 7.34·45-s + 4.85·47-s − 3.51·49-s − 9.36·51-s + 0.358·53-s − 4.16·55-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 1.60·5-s + 0.705·7-s + 0.681·9-s − 0.349·11-s + 0.541·13-s − 2.08·15-s + 1.01·17-s + 1.52·19-s − 0.915·21-s − 1.37·23-s + 1.58·25-s + 0.412·27-s + 0.907·29-s + 1.63·31-s + 0.453·33-s + 1.13·35-s + 0.778·37-s − 0.701·39-s − 0.0653·41-s − 1.53·43-s + 1.09·45-s + 0.708·47-s − 0.501·49-s − 1.31·51-s + 0.0492·53-s − 0.562·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.308173281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.308173281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 6.60T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 0.418T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 - 0.358T + 53T^{2} \) |
| 59 | \( 1 + 2.67T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 3.77T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 1.64T + 97T^{2} \) |
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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
−7.82515407218719746300325466506, −6.76711180426341268461509729861, −6.05124886862991820242086769741, −5.89492494980073763057301058027, −4.99379947473943250689928647539, −4.80833597860390880659166409728, −3.40875163010647899993164854398, −2.49150372666440369844555019885, −1.48270855939773961560513697706, −0.884611760974603181610831142536,
0.884611760974603181610831142536, 1.48270855939773961560513697706, 2.49150372666440369844555019885, 3.40875163010647899993164854398, 4.80833597860390880659166409728, 4.99379947473943250689928647539, 5.89492494980073763057301058027, 6.05124886862991820242086769741, 6.76711180426341268461509729861, 7.82515407218719746300325466506
