| L(s) = 1 | − 2-s − 4-s + 4·5-s − 2·7-s + 3·8-s − 4·10-s + 11-s + 2·13-s + 2·14-s − 16-s − 2·17-s − 4·20-s − 22-s − 4·23-s + 11·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 5·32-s + 2·34-s − 8·35-s + 6·37-s + 12·40-s − 10·41-s + 6·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.755·7-s + 1.06·8-s − 1.26·10-s + 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.894·20-s − 0.213·22-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 1.35·35-s + 0.986·37-s + 1.89·40-s − 1.56·41-s + 0.914·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.10513151015001, −14.53842216108540, −13.97077829562711, −13.56988737950937, −13.22520355104426, −12.77893306857432, −12.18212297361010, −11.19083820732738, −10.79772581848862, −10.16844504183719, −9.713852666267388, −9.472970785371913, −8.768609309523580, −8.617080873975320, −7.561042592122057, −7.106616181730193, −6.233906106378191, −6.011467114136856, −5.381734867369248, −4.656652411778674, −3.934476019348819, −3.238925169084478, −2.267688698910028, −1.772284425708589, −1.032749093783487, 0,
1.032749093783487, 1.772284425708589, 2.267688698910028, 3.238925169084478, 3.934476019348819, 4.656652411778674, 5.381734867369248, 6.011467114136856, 6.233906106378191, 7.106616181730193, 7.561042592122057, 8.617080873975320, 8.768609309523580, 9.472970785371913, 9.713852666267388, 10.16844504183719, 10.79772581848862, 11.19083820732738, 12.18212297361010, 12.77893306857432, 13.22520355104426, 13.56988737950937, 13.97077829562711, 14.53842216108540, 15.10513151015001
