| L(s) = 1 | − 3-s − 3·5-s − 7-s − 2·9-s − 13-s + 3·15-s + 3·17-s + 2·19-s + 21-s + 4·25-s + 5·27-s − 6·29-s + 4·31-s + 3·35-s − 7·37-s + 39-s − 43-s + 6·45-s − 3·47-s − 6·49-s − 3·51-s − 2·57-s + 6·59-s − 8·61-s + 2·63-s + 3·65-s − 14·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.277·13-s + 0.774·15-s + 0.727·17-s + 0.458·19-s + 0.218·21-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 0.160·39-s − 0.152·43-s + 0.894·45-s − 0.437·47-s − 6/7·49-s − 0.420·51-s − 0.264·57-s + 0.781·59-s − 1.02·61-s + 0.251·63-s + 0.372·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.59290438264090, −15.21579290554112, −14.63698574441739, −14.09032332780271, −13.48558132345938, −12.73208959082445, −12.27424643530977, −11.69578267506747, −11.57867887126830, −10.81315097528699, −10.33380762509325, −9.624596986057958, −8.995313410918769, −8.358275273382934, −7.802132683246224, −7.354945934563506, −6.662247135519905, −6.037379983299525, −5.358544083691704, −4.836972014758551, −4.080085447597959, −3.316361643734898, −3.033195438867576, −1.834678925543452, −0.7148485006380773, 0,
0.7148485006380773, 1.834678925543452, 3.033195438867576, 3.316361643734898, 4.080085447597959, 4.836972014758551, 5.358544083691704, 6.037379983299525, 6.662247135519905, 7.354945934563506, 7.802132683246224, 8.358275273382934, 8.995313410918769, 9.624596986057958, 10.33380762509325, 10.81315097528699, 11.57867887126830, 11.69578267506747, 12.27424643530977, 12.73208959082445, 13.48558132345938, 14.09032332780271, 14.63698574441739, 15.21579290554112, 15.59290438264090
