| L(s) = 1 | − 2-s + 10·3-s − 7·4-s − 10·6-s − 23·7-s + 15·8-s + 73·9-s − 43·11-s − 70·12-s + 79·13-s + 23·14-s + 41·16-s + 96·17-s − 73·18-s − 103·19-s − 230·21-s + 43·22-s + 23·23-s + 150·24-s − 79·26-s + 460·27-s + 161·28-s − 27·29-s + 110·31-s − 161·32-s − 430·33-s − 96·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.92·3-s − 7/8·4-s − 0.680·6-s − 1.24·7-s + 0.662·8-s + 2.70·9-s − 1.17·11-s − 1.68·12-s + 1.68·13-s + 0.439·14-s + 0.640·16-s + 1.36·17-s − 0.955·18-s − 1.24·19-s − 2.39·21-s + 0.416·22-s + 0.208·23-s + 1.27·24-s − 0.595·26-s + 3.27·27-s + 1.08·28-s − 0.172·29-s + 0.637·31-s − 0.889·32-s − 2.26·33-s − 0.484·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.549233358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549233358\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 7 | \( 1 + 23 T + p^{3} T^{2} \) |
| 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 - 79 T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 103 T + p^{3} T^{2} \) |
| 29 | \( 1 + 27 T + p^{3} T^{2} \) |
| 31 | \( 1 - 110 T + p^{3} T^{2} \) |
| 37 | \( 1 - 42 T + p^{3} T^{2} \) |
| 41 | \( 1 - 171 T + p^{3} T^{2} \) |
| 43 | \( 1 - 265 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 428 T + p^{3} T^{2} \) |
| 59 | \( 1 - 638 T + p^{3} T^{2} \) |
| 61 | \( 1 - 842 T + p^{3} T^{2} \) |
| 67 | \( 1 - 80 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1050 T + p^{3} T^{2} \) |
| 73 | \( 1 + 87 T + p^{3} T^{2} \) |
| 79 | \( 1 - 17 T + p^{3} T^{2} \) |
| 83 | \( 1 + 63 T + p^{3} T^{2} \) |
| 89 | \( 1 - 126 T + p^{3} T^{2} \) |
| 97 | \( 1 + 668 T + p^{3} T^{2} \) |
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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
−10.07499575301733685599316401305, −9.286568606640605975755270091999, −8.541290351217952124851656314250, −8.086504832817550041864334694664, −7.11250490439810228338390315434, −5.76312039949662317849780474199, −4.16079682545532855811287047965, −3.52191151641414717570642247557, −2.55698179576936476238774193383, −0.972151257785625839077143670384,
0.972151257785625839077143670384, 2.55698179576936476238774193383, 3.52191151641414717570642247557, 4.16079682545532855811287047965, 5.76312039949662317849780474199, 7.11250490439810228338390315434, 8.086504832817550041864334694664, 8.541290351217952124851656314250, 9.286568606640605975755270091999, 10.07499575301733685599316401305
