| L(s) = 1 | + (−2.81 + 0.113i)2-s + (0.467 + 1.61i)3-s + (5.93 − 0.479i)4-s + (1.59 − 0.605i)5-s + (−1.50 − 4.49i)6-s + (1.15 − 2.71i)7-s + (−11.0 + 1.34i)8-s + (0.147 − 0.0930i)9-s + (−4.43 + 1.88i)10-s + (1.00 − 1.58i)11-s + (3.55 + 9.36i)12-s + (−1.18 − 3.40i)13-s + (−2.95 + 7.78i)14-s + (1.72 + 2.29i)15-s + (19.3 − 3.13i)16-s + (−0.139 − 0.0595i)17-s + ⋯ |
| L(s) = 1 | + (−1.99 + 0.0803i)2-s + (0.270 + 0.932i)3-s + (2.96 − 0.239i)4-s + (0.714 − 0.270i)5-s + (−0.612 − 1.83i)6-s + (0.437 − 1.02i)7-s + (−3.91 + 0.475i)8-s + (0.0490 − 0.0310i)9-s + (−1.40 + 0.597i)10-s + (0.302 − 0.477i)11-s + (1.02 + 2.70i)12-s + (−0.329 − 0.944i)13-s + (−0.788 + 2.08i)14-s + (0.445 + 0.592i)15-s + (4.82 − 0.784i)16-s + (−0.0338 − 0.0144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.650002 + 0.0429729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.650002 + 0.0429729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.18 + 3.40i)T \) |
| good | 2 | \( 1 + (2.81 - 0.113i)T + (1.99 - 0.160i)T^{2} \) |
| 3 | \( 1 + (-0.467 - 1.61i)T + (-2.53 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-1.59 + 0.605i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (-1.15 + 2.71i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 1.58i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (0.139 + 0.0595i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (-3.60 - 2.08i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.549 + 0.951i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.145 - 3.60i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (3.40 - 3.84i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (3.51 + 0.717i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (4.62 - 1.34i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (-0.658 - 3.22i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (-7.83 - 5.40i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (0.943 + 7.76i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.889 + 5.47i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (8.26 + 6.21i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (1.16 - 14.4i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (3.77 + 3.62i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (0.450 + 0.857i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.37i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-2.87 - 11.6i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (-4.63 + 2.67i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.11 - 4.99i)T + (19.4 - 95.0i)T^{2} \) |
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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
−12.44373009086945728267557515072, −11.10126590250307825760340306237, −10.39169621825813320957257950888, −9.777747466113841002943041045894, −8.981176390255912437388051659011, −7.939861154170715161821584247225, −6.95036325655286188234252953919, −5.50423351128108344195760150065, −3.31759687324944794412165861586, −1.31598885904049388558152219290,
1.74524081161765026228300052735, 2.42038113875547174657793257599, 5.90792126818545123972924995665, 6.95481225799952250236802754420, 7.67034193398040676638364435235, 8.826779852648619347797072115989, 9.477202908702137832203874224508, 10.46491752486020040012959290559, 11.80908103752512245586135486808, 12.12434204328674350757886720327
