| L(s) = 1 | + 3·2-s − 3·3-s + 4-s − 5·5-s − 9·6-s + 20·7-s − 21·8-s + 9·9-s − 15·10-s − 24·11-s − 3·12-s + 74·13-s + 60·14-s + 15·15-s − 71·16-s + 54·17-s + 27·18-s − 124·19-s − 5·20-s − 60·21-s − 72·22-s − 120·23-s + 63·24-s + 25·25-s + 222·26-s − 27·27-s + 20·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 1/8·4-s − 0.447·5-s − 0.612·6-s + 1.07·7-s − 0.928·8-s + 1/3·9-s − 0.474·10-s − 0.657·11-s − 0.0721·12-s + 1.57·13-s + 1.14·14-s + 0.258·15-s − 1.10·16-s + 0.770·17-s + 0.353·18-s − 1.49·19-s − 0.0559·20-s − 0.623·21-s − 0.697·22-s − 1.08·23-s + 0.535·24-s + 1/5·25-s + 1.67·26-s − 0.192·27-s + 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.253715643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253715643\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−18.70300043193769683752896167074, −17.69200804100244789944050020873, −15.95630405612152238344786357291, −14.75142913842011302176145084551, −13.41204716467712329166424680568, −12.07947160584376184757399764537, −10.84203000361767538734049526530, −8.293604804757399339597796518127, −5.87964458350579542005265935064, −4.25806719846960648281623011472,
4.25806719846960648281623011472, 5.87964458350579542005265935064, 8.293604804757399339597796518127, 10.84203000361767538734049526530, 12.07947160584376184757399764537, 13.41204716467712329166424680568, 14.75142913842011302176145084551, 15.95630405612152238344786357291, 17.69200804100244789944050020873, 18.70300043193769683752896167074
