| L(s) = 1 | − 3-s + 9-s − 4·11-s − 7·13-s − 6·17-s − 3·19-s − 2·23-s − 27-s + 2·29-s + 7·31-s + 4·33-s − 7·37-s + 7·39-s + 8·41-s − 5·43-s − 10·47-s + 6·51-s − 8·53-s + 3·57-s − 10·59-s − 6·61-s − 3·67-s + 2·69-s + 15·73-s − 79-s + 81-s + 8·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.94·13-s − 1.45·17-s − 0.688·19-s − 0.417·23-s − 0.192·27-s + 0.371·29-s + 1.25·31-s + 0.696·33-s − 1.15·37-s + 1.12·39-s + 1.24·41-s − 0.762·43-s − 1.45·47-s + 0.840·51-s − 1.09·53-s + 0.397·57-s − 1.30·59-s − 0.768·61-s − 0.366·67-s + 0.240·69-s + 1.75·73-s − 0.112·79-s + 1/9·81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.91626776364226, −12.69664534067497, −12.24007141749717, −11.83087139845668, −11.22393670250426, −10.81858698700529, −10.37116957850705, −9.982463259530260, −9.522557486340567, −8.995111306059088, −8.358765943527531, −7.837919847886862, −7.577968814761837, −6.856519888604934, −6.459292965346247, −6.116630456772429, −5.167893494533958, −4.985226082338963, −4.594236700259470, −4.058491684725735, −3.120587647367707, −2.638388013194666, −2.164474290079300, −1.596732594856940, −0.4353887830318852, 0,
0.4353887830318852, 1.596732594856940, 2.164474290079300, 2.638388013194666, 3.120587647367707, 4.058491684725735, 4.594236700259470, 4.985226082338963, 5.167893494533958, 6.116630456772429, 6.459292965346247, 6.856519888604934, 7.577968814761837, 7.837919847886862, 8.358765943527531, 8.995111306059088, 9.522557486340567, 9.982463259530260, 10.37116957850705, 10.81858698700529, 11.22393670250426, 11.83087139845668, 12.24007141749717, 12.69664534067497, 12.91626776364226
