| L(s) = 1 | + (0.0371 + 0.138i)2-s + (0.407 + 0.109i)3-s + (1.71 − 0.989i)4-s + (2.06 + 2.06i)5-s + 0.0605i·6-s + (0.858 − 0.659i)7-s + (0.403 + 0.403i)8-s + (−2.44 − 1.41i)9-s + (−0.209 + 0.362i)10-s + (−2.67 + 2.05i)11-s + (0.807 − 0.216i)12-s + (−1.13 − 0.873i)13-s + (0.123 + 0.0945i)14-s + (0.616 + 1.06i)15-s + (1.93 − 3.35i)16-s + (−1.09 + 0.453i)17-s + ⋯ |
| L(s) = 1 | + (0.0262 + 0.0979i)2-s + (0.235 + 0.0630i)3-s + (0.857 − 0.494i)4-s + (0.924 + 0.924i)5-s + 0.0247i·6-s + (0.324 − 0.249i)7-s + (0.142 + 0.142i)8-s + (−0.814 − 0.470i)9-s + (−0.0662 + 0.114i)10-s + (−0.807 + 0.619i)11-s + (0.232 − 0.0624i)12-s + (−0.315 − 0.242i)13-s + (0.0329 + 0.0252i)14-s + (0.159 + 0.275i)15-s + (0.484 − 0.839i)16-s + (−0.265 + 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70069 + 0.170501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70069 + 0.170501i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 + (6.88 + 13.9i)T \) |
| good | 2 | \( 1 + (-0.0371 - 0.138i)T + (-1.73 + i)T^{2} \) |
| 3 | \( 1 + (-0.407 - 0.109i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-2.06 - 2.06i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.858 + 0.659i)T + (1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (2.67 - 2.05i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.13 + 0.873i)T + (3.36 + 12.5i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.453i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.756 - 0.985i)T + (-4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 1.00i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.13 - 0.304i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.800 + 6.08i)T + (-29.9 - 8.02i)T^{2} \) |
| 37 | \( 1 + (3.68 - 2.82i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (5.36 - 5.36i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.0807 + 0.194i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (7.38 + 7.38i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.65 - 2.58i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.06 + 7.69i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.73 + 1.73i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.152 + 0.569i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.110 - 0.842i)T + (-68.5 - 18.3i)T^{2} \) |
| 73 | \( 1 + (8.56 + 3.54i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.05 + 2.05i)T - 79iT^{2} \) |
| 83 | \( 1 + (-7.30 - 4.21i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.97 - 15.0i)T + (-85.9 - 23.0i)T^{2} \) |
| 97 | \( 1 + (-0.317 + 0.183i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.94112045343738925682351992764, −11.04588808843880303950845657300, −10.26485695616045465796869247847, −9.576134247629085161499463972243, −8.046838917629512614232114380755, −7.00899525126674542000529170009, −6.13241179027797392206178184221, −5.15891688494178499531367171678, −3.06614518213146533443072493722, −2.07997053155394753573589876110,
1.92586057324824964427454226948, 3.02228177589033747067890083658, 4.99987429352215177970177768118, 5.80997039071034631448423575415, 7.17633233048576980973162715843, 8.364731402464958259720918873626, 8.908586060683930717846824942265, 10.30676200662530570481862463725, 11.20015915054090417180768877253, 12.09609883358452356266316443426
