| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 4·11-s − 12-s + 15-s + 16-s − 4·17-s + 18-s − 4·19-s − 20-s + 4·22-s − 6·23-s − 24-s + 25-s − 27-s + 4·29-s + 30-s − 10·31-s + 32-s − 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.182·30-s − 1.79·31-s + 0.176·32-s − 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.65983681102560675700445480667, −6.95053828073891001384807284791, −6.33595104718497942044298295964, −5.80157446091064440946839447805, −4.78242823547330893025968019484, −4.14332597475558007058504278260, −3.67073593988599946136413200387, −2.39107645053797039693464571703, −1.47574214305438027301467539464, 0,
1.47574214305438027301467539464, 2.39107645053797039693464571703, 3.67073593988599946136413200387, 4.14332597475558007058504278260, 4.78242823547330893025968019484, 5.80157446091064440946839447805, 6.33595104718497942044298295964, 6.95053828073891001384807284791, 7.65983681102560675700445480667
