| L(s) = 1 | + 4.93·2-s + 2.28·3-s + 16.3·4-s − 5·5-s + 11.2·6-s + 15.1·7-s + 41.4·8-s − 21.7·9-s − 24.6·10-s − 63.7·11-s + 37.3·12-s − 13·13-s + 74.9·14-s − 11.4·15-s + 73.6·16-s − 43.6·17-s − 107.·18-s − 1.25·19-s − 81.9·20-s + 34.6·21-s − 314.·22-s − 16.0·23-s + 94.5·24-s + 25·25-s − 64.2·26-s − 111.·27-s + 248.·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.438·3-s + 2.04·4-s − 0.447·5-s + 0.766·6-s + 0.819·7-s + 1.83·8-s − 0.807·9-s − 0.780·10-s − 1.74·11-s + 0.899·12-s − 0.277·13-s + 1.43·14-s − 0.196·15-s + 1.15·16-s − 0.622·17-s − 1.40·18-s − 0.0151·19-s − 0.916·20-s + 0.359·21-s − 3.05·22-s − 0.145·23-s + 0.804·24-s + 0.200·25-s − 0.484·26-s − 0.793·27-s + 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 4.93T + 8T^{2} \) |
| 3 | \( 1 - 2.28T + 27T^{2} \) |
| 7 | \( 1 - 15.1T + 343T^{2} \) |
| 11 | \( 1 + 63.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.25T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 161.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 684.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 713.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 419.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 455.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 243.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 786.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 277.T + 9.12e5T^{2} \) |
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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
−7.981314944774748550045276505330, −7.74361293116859481039809099675, −6.57175027917520121233350564214, −5.77170144374629687376566369089, −4.82878821739487278240102955125, −4.64486622226449527031371730393, −3.29508699427378845771216471618, −2.77361951635821828097487204019, −1.92303236286842971579754140752, 0,
1.92303236286842971579754140752, 2.77361951635821828097487204019, 3.29508699427378845771216471618, 4.64486622226449527031371730393, 4.82878821739487278240102955125, 5.77170144374629687376566369089, 6.57175027917520121233350564214, 7.74361293116859481039809099675, 7.981314944774748550045276505330
