| L(s) = 1 | − 4.53·3-s − 20.8·5-s + 16.7·7-s − 6.40·9-s − 72.1·11-s − 13·13-s + 94.4·15-s − 21.2·17-s − 66.7·19-s − 76.2·21-s − 149.·23-s + 307.·25-s + 151.·27-s − 81.0·29-s − 177.·31-s + 327.·33-s − 349.·35-s + 353.·37-s + 58.9·39-s + 284.·41-s + 31.8·43-s + 133.·45-s + 112.·47-s − 60.8·49-s + 96.2·51-s + 211.·53-s + 1.50e3·55-s + ⋯ |
| L(s) = 1 | − 0.873·3-s − 1.86·5-s + 0.906·7-s − 0.237·9-s − 1.97·11-s − 0.277·13-s + 1.62·15-s − 0.302·17-s − 0.805·19-s − 0.792·21-s − 1.35·23-s + 2.46·25-s + 1.08·27-s − 0.518·29-s − 1.02·31-s + 1.72·33-s − 1.68·35-s + 1.56·37-s + 0.242·39-s + 1.08·41-s + 0.112·43-s + 0.441·45-s + 0.347·47-s − 0.177·49-s + 0.264·51-s + 0.548·53-s + 3.68·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2996853885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2996853885\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 4.53T + 27T^{2} \) |
| 5 | \( 1 + 20.8T + 125T^{2} \) |
| 7 | \( 1 - 16.7T + 343T^{2} \) |
| 11 | \( 1 + 72.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 21.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 284.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 112.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 227.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 723.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 15.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 972.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 155.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 517.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
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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
−11.12292003677402108923233324674, −10.32574826074189033302778138425, −8.589698991829085163528727467016, −7.913860149635067133960156269906, −7.36674052367949654123194237368, −5.84778493771687643946400343738, −4.89758208120329195167767685419, −4.09914408337403873325395980399, −2.55806409216120429423647926168, −0.34756446319607860798650849152,
0.34756446319607860798650849152, 2.55806409216120429423647926168, 4.09914408337403873325395980399, 4.89758208120329195167767685419, 5.84778493771687643946400343738, 7.36674052367949654123194237368, 7.913860149635067133960156269906, 8.589698991829085163528727467016, 10.32574826074189033302778138425, 11.12292003677402108923233324674
