L(s) = 1 | − 2·2-s − 3-s + 4·4-s − 5·5-s + 2·6-s + 7·7-s − 8·8-s − 26·9-s + 10·10-s − 65·11-s − 4·12-s + 13·13-s − 14·14-s + 5·15-s + 16·16-s − 73·17-s + 52·18-s − 142·19-s − 20·20-s − 7·21-s + 130·22-s + 130·23-s + 8·24-s + 25·25-s − 26·26-s + 53·27-s + 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s + 0.377·7-s − 0.353·8-s − 0.962·9-s + 0.316·10-s − 1.78·11-s − 0.0962·12-s + 0.277·13-s − 0.267·14-s + 0.0860·15-s + 1/4·16-s − 1.04·17-s + 0.680·18-s − 1.71·19-s − 0.223·20-s − 0.0727·21-s + 1.25·22-s + 1.17·23-s + 0.0680·24-s + 1/5·25-s − 0.196·26-s + 0.377·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 65 T + p^{3} T^{2} \) |
| 13 | \( 1 - p T + p^{3} T^{2} \) |
| 17 | \( 1 + 73 T + p^{3} T^{2} \) |
| 19 | \( 1 + 142 T + p^{3} T^{2} \) |
| 23 | \( 1 - 130 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 424 T + p^{3} T^{2} \) |
| 43 | \( 1 - 534 T + p^{3} T^{2} \) |
| 47 | \( 1 + 269 T + p^{3} T^{2} \) |
| 53 | \( 1 + 132 T + p^{3} T^{2} \) |
| 59 | \( 1 + 224 T + p^{3} T^{2} \) |
| 61 | \( 1 + 572 T + p^{3} T^{2} \) |
| 67 | \( 1 + 108 T + p^{3} T^{2} \) |
| 71 | \( 1 - 560 T + p^{3} T^{2} \) |
| 73 | \( 1 - 586 T + p^{3} T^{2} \) |
| 79 | \( 1 - 57 T + p^{3} T^{2} \) |
| 83 | \( 1 - 252 T + p^{3} T^{2} \) |
| 89 | \( 1 + 184 T + p^{3} T^{2} \) |
| 97 | \( 1 + 605 T + p^{3} T^{2} \) |
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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
−13.60842689029208891484136047206, −12.39052946823286851782757737121, −11.05410024323683362654608201773, −10.57010912787466722829606692939, −8.725528656085196856102093917550, −8.068976943693021265184963729001, −6.53347381492851810807231914084, −4.93486165220516524173846325560, −2.64171127784627348247258169356, 0,
2.64171127784627348247258169356, 4.93486165220516524173846325560, 6.53347381492851810807231914084, 8.068976943693021265184963729001, 8.725528656085196856102093917550, 10.57010912787466722829606692939, 11.05410024323683362654608201773, 12.39052946823286851782757737121, 13.60842689029208891484136047206