| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 13-s + 2·14-s − 15-s + 16-s − 8·17-s + 18-s − 6·19-s − 20-s + 2·21-s − 4·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.90348802474532, −12.38919837083781, −11.81519217382564, −11.30170872252796, −11.02925834670458, −10.57301274097888, −10.13575133689349, −9.648389360328411, −8.756168632765812, −8.558548859203895, −8.156748237383484, −7.835077284228965, −7.028754234715554, −6.863029979538796, −6.218540053831079, −5.689996291829926, −5.121042006831642, −4.572635951365377, −4.195520408284698, −3.955084244352945, −3.134586888321385, −2.473310484395371, −2.219191295678747, −1.773867216771024, −0.7513229242439229, 0,
0.7513229242439229, 1.773867216771024, 2.219191295678747, 2.473310484395371, 3.134586888321385, 3.955084244352945, 4.195520408284698, 4.572635951365377, 5.121042006831642, 5.689996291829926, 6.218540053831079, 6.863029979538796, 7.028754234715554, 7.835077284228965, 8.156748237383484, 8.558548859203895, 8.756168632765812, 9.648389360328411, 10.13575133689349, 10.57301274097888, 11.02925834670458, 11.30170872252796, 11.81519217382564, 12.38919837083781, 12.90348802474532
