L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s − 4·13-s − 16-s + 3·18-s + 4·19-s + 20-s − 6·23-s + 25-s + 4·26-s − 6·31-s − 5·32-s + 3·36-s + 2·37-s − 4·38-s − 3·40-s − 2·41-s − 12·43-s + 3·45-s + 6·46-s − 8·47-s − 50-s + 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s − 1.10·13-s − 1/4·16-s + 0.707·18-s + 0.917·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.784·26-s − 1.07·31-s − 0.883·32-s + 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s − 1.82·43-s + 0.447·45-s + 0.884·46-s − 1.16·47-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.90018835779264, −14.99423293002819, −14.60990369756792, −14.19648597976459, −13.58448472033072, −13.13105464488524, −12.20465572857485, −12.11361153849788, −11.20003206771007, −10.99241097452040, −10.06206066939690, −9.646383204541294, −9.359621460486729, −8.420800476381803, −8.186612955035387, −7.677404478819901, −7.034591644785657, −6.362148047290545, −5.399405954425043, −5.134503067959207, −4.397932948797195, −3.625289779109346, −3.052200726369852, −2.121224655987091, −1.335030249677101, 0, 0,
1.335030249677101, 2.121224655987091, 3.052200726369852, 3.625289779109346, 4.397932948797195, 5.134503067959207, 5.399405954425043, 6.362148047290545, 7.034591644785657, 7.677404478819901, 8.186612955035387, 8.420800476381803, 9.359621460486729, 9.646383204541294, 10.06206066939690, 10.99241097452040, 11.20003206771007, 12.11361153849788, 12.20465572857485, 13.13105464488524, 13.58448472033072, 14.19648597976459, 14.60990369756792, 14.99423293002819, 15.90018835779264