| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 8-s − 2·9-s + 4·10-s − 5·11-s − 12-s − 4·13-s + 4·15-s + 16-s − 17-s + 2·18-s + 6·19-s − 4·20-s + 5·22-s − 3·23-s + 24-s + 11·25-s + 4·26-s + 5·27-s + 29-s − 4·30-s + 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.50·11-s − 0.288·12-s − 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.471·18-s + 1.37·19-s − 0.894·20-s + 1.06·22-s − 0.625·23-s + 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.962·27-s + 0.185·29-s − 0.730·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 223 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.66834866008401, −15.15977206344125, −14.57697460643521, −14.02107885622805, −13.24323668370103, −12.50951353055371, −12.09428320517540, −11.78827455226030, −11.23176710371141, −10.77946276829290, −10.29393926308080, −9.631343173577185, −8.994024574654918, −8.229474951937062, −7.802940492708459, −7.680738521815433, −6.899830964847408, −6.319028063953004, −5.338321586301648, −5.051970894752380, −4.409235830939632, −3.404232694054828, −2.974868462565870, −2.313436075257054, −1.017055272059934, 0, 0,
1.017055272059934, 2.313436075257054, 2.974868462565870, 3.404232694054828, 4.409235830939632, 5.051970894752380, 5.338321586301648, 6.319028063953004, 6.899830964847408, 7.680738521815433, 7.802940492708459, 8.229474951937062, 8.994024574654918, 9.631343173577185, 10.29393926308080, 10.77946276829290, 11.23176710371141, 11.78827455226030, 12.09428320517540, 12.50951353055371, 13.24323668370103, 14.02107885622805, 14.57697460643521, 15.15977206344125, 15.66834866008401
