| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·11-s + 4·13-s + 2·15-s + 17-s − 2·19-s + 6·23-s − 25-s − 27-s − 8·29-s + 4·31-s + 2·33-s + 6·37-s − 4·39-s − 6·41-s − 4·43-s − 2·45-s − 4·47-s − 51-s − 2·53-s + 4·55-s + 2·57-s + 4·59-s − 10·61-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.516·15-s + 0.242·17-s − 0.458·19-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s − 0.140·51-s − 0.274·53-s + 0.539·55-s + 0.264·57-s + 0.520·59-s − 1.28·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.30102409734481, −12.98727392437047, −12.71601943397518, −11.85269348823714, −11.65616750805874, −11.12168397510719, −10.87764993313423, −10.25671857958893, −9.794488405012353, −9.172484289248844, −8.566621653568055, −8.258775585166162, −7.606929398113164, −7.294125795763448, −6.662633679238566, −6.108055400361449, −5.679374835832997, −5.080011769114114, −4.485402028028706, −4.110171678893955, −3.273778866877765, −3.143791072477835, −2.072770314271676, −1.451350688662722, −0.6910834540685544, 0,
0.6910834540685544, 1.451350688662722, 2.072770314271676, 3.143791072477835, 3.273778866877765, 4.110171678893955, 4.485402028028706, 5.080011769114114, 5.679374835832997, 6.108055400361449, 6.662633679238566, 7.294125795763448, 7.606929398113164, 8.258775585166162, 8.566621653568055, 9.172484289248844, 9.794488405012353, 10.25671857958893, 10.87764993313423, 11.12168397510719, 11.65616750805874, 11.85269348823714, 12.71601943397518, 12.98727392437047, 13.30102409734481
