| L(s) = 1 | + 3.48·2-s + 3·3-s + 4.17·4-s − 0.768·5-s + 10.4·6-s − 13.3·8-s + 9·9-s − 2.68·10-s − 11·11-s + 12.5·12-s − 37.4·13-s − 2.30·15-s − 79.9·16-s + 22.4·17-s + 31.3·18-s + 74.1·19-s − 3.20·20-s − 38.3·22-s + 195.·23-s − 40.0·24-s − 124.·25-s − 130.·26-s + 27·27-s + 48.3·29-s − 8.04·30-s − 287.·31-s − 172.·32-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.577·3-s + 0.521·4-s − 0.0687·5-s + 0.712·6-s − 0.590·8-s + 0.333·9-s − 0.0847·10-s − 0.301·11-s + 0.300·12-s − 0.799·13-s − 0.0396·15-s − 1.24·16-s + 0.320·17-s + 0.411·18-s + 0.895·19-s − 0.0358·20-s − 0.371·22-s + 1.76·23-s − 0.340·24-s − 0.995·25-s − 0.986·26-s + 0.192·27-s + 0.309·29-s − 0.0489·30-s − 1.66·31-s − 0.950·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.48T + 8T^{2} \) |
| 5 | \( 1 + 0.768T + 125T^{2} \) |
| 13 | \( 1 + 37.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 329.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 569.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 869.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 942.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 320.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 963.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 53.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−8.756650818361657454762460483787, −7.56051348983616372602421670874, −7.09481031513293895543884869915, −5.92067611183062213303253920375, −5.14984364685240116691842716228, −4.52421080616988962861303372639, −3.37517640635140272310820532651, −2.96149997954781925802647421701, −1.69223288140630265812525974921, 0,
1.69223288140630265812525974921, 2.96149997954781925802647421701, 3.37517640635140272310820532651, 4.52421080616988962861303372639, 5.14984364685240116691842716228, 5.92067611183062213303253920375, 7.09481031513293895543884869915, 7.56051348983616372602421670874, 8.756650818361657454762460483787
