| L(s) = 1 | + 2-s − 7·3-s − 7·4-s − 7·6-s − 15·8-s + 22·9-s − 43·11-s + 49·12-s + 28·13-s + 41·16-s − 91·17-s + 22·18-s + 35·19-s − 43·22-s + 162·23-s + 105·24-s + 28·26-s + 35·27-s + 160·29-s − 42·31-s + 161·32-s + 301·33-s − 91·34-s − 154·36-s − 314·37-s + 35·38-s − 196·39-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.34·3-s − 7/8·4-s − 0.476·6-s − 0.662·8-s + 0.814·9-s − 1.17·11-s + 1.17·12-s + 0.597·13-s + 0.640·16-s − 1.29·17-s + 0.288·18-s + 0.422·19-s − 0.416·22-s + 1.46·23-s + 0.893·24-s + 0.211·26-s + 0.249·27-s + 1.02·29-s − 0.243·31-s + 0.889·32-s + 1.58·33-s − 0.459·34-s − 0.712·36-s − 1.39·37-s + 0.149·38-s − 0.804·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 91 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 160 T + p^{3} T^{2} \) |
| 31 | \( 1 + 42 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 203 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 196 T + p^{3} T^{2} \) |
| 53 | \( 1 - 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 280 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 141 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 + 777 T + p^{3} T^{2} \) |
| 89 | \( 1 - 945 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1246 T + p^{3} T^{2} \) |
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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
−8.914809895094261190331668071022, −8.246560856316936871129274975654, −7.00716148536209536910540526199, −6.26670559017336053728345099836, −5.16553865065355082646641963757, −5.09696743634974546613535499900, −3.93827318184214908039349194751, −2.72443318535649633635183444967, −0.940139905114092155886518375631, 0,
0.940139905114092155886518375631, 2.72443318535649633635183444967, 3.93827318184214908039349194751, 5.09696743634974546613535499900, 5.16553865065355082646641963757, 6.26670559017336053728345099836, 7.00716148536209536910540526199, 8.246560856316936871129274975654, 8.914809895094261190331668071022
