| L(s) = 1 | − 3·3-s − 2·5-s + 7·7-s + 9·9-s − 52·11-s + 86·13-s + 6·15-s − 30·17-s + 4·19-s − 21·21-s − 120·23-s − 121·25-s − 27·27-s + 246·29-s − 80·31-s + 156·33-s − 14·35-s − 290·37-s − 258·39-s − 374·41-s − 164·43-s − 18·45-s − 464·47-s + 49·49-s + 90·51-s − 162·53-s + 104·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.178·5-s + 0.377·7-s + 1/3·9-s − 1.42·11-s + 1.83·13-s + 0.103·15-s − 0.428·17-s + 0.0482·19-s − 0.218·21-s − 1.08·23-s − 0.967·25-s − 0.192·27-s + 1.57·29-s − 0.463·31-s + 0.822·33-s − 0.0676·35-s − 1.28·37-s − 1.05·39-s − 1.42·41-s − 0.581·43-s − 0.0596·45-s − 1.44·47-s + 1/7·49-s + 0.247·51-s − 0.419·53-s + 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 290 T + p^{3} T^{2} \) |
| 41 | \( 1 + 374 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 666 T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 + 518 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 220 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
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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
−10.72587527786047212164482629811, −10.02500572128820569526497888800, −8.527894033764317019662339860853, −7.975158718402493533231181536283, −6.64113259394980296856508710088, −5.71694470682103014271866949403, −4.69790005682046383204412999804, −3.42320419010303618354513636155, −1.70794581072032516281605836139, 0,
1.70794581072032516281605836139, 3.42320419010303618354513636155, 4.69790005682046383204412999804, 5.71694470682103014271866949403, 6.64113259394980296856508710088, 7.975158718402493533231181536283, 8.527894033764317019662339860853, 10.02500572128820569526497888800, 10.72587527786047212164482629811
