L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s − 2·14-s − 15-s + 16-s − 2·17-s + 18-s − 20-s − 2·21-s − 2·23-s + 24-s + 25-s − 2·26-s + 27-s − 2·28-s + 4·29-s − 30-s + 4·31-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 + 0.288·12-s − 0.554·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.64958765318396, −15.93409588731427, −15.64634516925209, −15.15815687156379, −14.43836163739419, −13.93875282259751, −13.48574664821230, −12.75018308537213, −12.32989544974756, −11.86306598161525, −11.01872493579644, −10.45342749163034, −9.792926257371640, −9.197821194150904, −8.480815611115517, −7.799479773384784, −7.216664133383383, −6.552468213261359, −6.004378343046248, −5.045603615568553, −4.381389455191669, −3.811887938382089, −2.936835113298251, −2.538897995190344, −1.385691988967204, 0,
1.385691988967204, 2.538897995190344, 2.936835113298251, 3.811887938382089, 4.381389455191669, 5.045603615568553, 6.004378343046248, 6.552468213261359, 7.216664133383383, 7.799479773384784, 8.480815611115517, 9.197821194150904, 9.792926257371640, 10.45342749163034, 11.01872493579644, 11.86306598161525, 12.32989544974756, 12.75018308537213, 13.48574664821230, 13.93875282259751, 14.43836163739419, 15.15815687156379, 15.64634516925209, 15.93409588731427, 16.64958765318396