| L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·11-s + 13-s − 6·17-s − 6·19-s − 2·21-s − 5·25-s + 27-s − 2·29-s + 6·31-s − 4·33-s + 10·37-s + 39-s + 8·41-s − 12·43-s − 12·47-s − 3·49-s − 6·51-s − 6·53-s − 6·57-s + 2·61-s − 2·63-s − 2·67-s + 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 1.24·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.794·57-s + 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.380691865886402984746233056078, −8.326566062545647802308043507710, −7.935593700998113984022213479788, −6.68084707610599017210864105435, −6.18680019324680760976289140185, −4.87228388162956512694940641649, −4.02331688083948425946247925030, −2.89365370501913595809144605675, −2.07382717156596942564746274979, 0,
2.07382717156596942564746274979, 2.89365370501913595809144605675, 4.02331688083948425946247925030, 4.87228388162956512694940641649, 6.18680019324680760976289140185, 6.68084707610599017210864105435, 7.935593700998113984022213479788, 8.326566062545647802308043507710, 9.380691865886402984746233056078
