| L(s) = 1 | − 2-s − 4-s − 3·7-s + 3·8-s + 11-s + 13-s + 3·14-s − 16-s + 5·17-s − 8·19-s − 22-s − 26-s + 3·28-s − 29-s + 3·31-s − 5·32-s − 5·34-s − 8·37-s + 8·38-s + 2·41-s + 8·43-s − 44-s + 11·47-s + 2·49-s − 52-s + 11·53-s − 9·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.13·7-s + 1.06·8-s + 0.301·11-s + 0.277·13-s + 0.801·14-s − 1/4·16-s + 1.21·17-s − 1.83·19-s − 0.213·22-s − 0.196·26-s + 0.566·28-s − 0.185·29-s + 0.538·31-s − 0.883·32-s − 0.857·34-s − 1.31·37-s + 1.29·38-s + 0.312·41-s + 1.21·43-s − 0.150·44-s + 1.60·47-s + 2/7·49-s − 0.138·52-s + 1.51·53-s − 1.20·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.595548958433079264111082383073, −7.75045701336173575216035835061, −6.97657605969525487423689555649, −6.17461920786826571136904894561, −5.41041083636789022339868573391, −4.23789841277672921578270813834, −3.70316863758305409233055175695, −2.52033112680127838456556863144, −1.19323552599730437177032777876, 0,
1.19323552599730437177032777876, 2.52033112680127838456556863144, 3.70316863758305409233055175695, 4.23789841277672921578270813834, 5.41041083636789022339868573391, 6.17461920786826571136904894561, 6.97657605969525487423689555649, 7.75045701336173575216035835061, 8.595548958433079264111082383073
