| L(s) = 1 | + (−28.7 + 9.33i)2-s + (−796. − 578. i)3-s + (−2.57e3 + 1.87e3i)4-s + (−7.51e3 + 2.31e4i)5-s + (2.82e4 + 9.19e3i)6-s + (−9.10e4 − 1.25e5i)7-s + (1.29e5 − 1.77e5i)8-s + (1.35e5 + 4.16e5i)9-s − 7.35e5i·10-s + (1.76e6 + 8.06e4i)11-s + 3.13e6·12-s + (−3.63e6 + 1.18e6i)13-s + (3.78e6 + 2.75e6i)14-s + (1.93e7 − 1.40e7i)15-s + (1.97e6 − 6.07e6i)16-s + (1.05e7 + 3.42e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.449 + 0.145i)2-s + (−1.09 − 0.793i)3-s + (−0.628 + 0.456i)4-s + (−0.481 + 1.48i)5-s + (0.606 + 0.197i)6-s + (−0.773 − 1.06i)7-s + (0.493 − 0.678i)8-s + (0.254 + 0.783i)9-s − 0.735i·10-s + (0.998 + 0.0455i)11-s + 1.04·12-s + (−0.752 + 0.244i)13-s + (0.502 + 0.365i)14-s + (1.70 − 1.23i)15-s + (0.117 − 0.362i)16-s + (0.436 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.472774 - 0.0805932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472774 - 0.0805932i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.76e6 - 8.06e4i)T \) |
| good | 2 | \( 1 + (28.7 - 9.33i)T + (3.31e3 - 2.40e3i)T^{2} \) |
| 3 | \( 1 + (796. + 578. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (7.51e3 - 2.31e4i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (9.10e4 + 1.25e5i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (3.63e6 - 1.18e6i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (-1.05e7 - 3.42e6i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (1.48e7 - 2.04e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 - 1.43e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (3.75e8 + 5.16e8i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-1.54e8 - 4.74e8i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (-3.80e9 + 2.76e9i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (3.92e9 - 5.40e9i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 + 7.73e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-6.14e9 - 4.46e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (8.42e9 + 2.59e10i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (-5.38e10 + 3.90e10i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (-3.86e10 - 1.25e10i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 - 6.25e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-2.23e9 + 6.86e9i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (-1.88e10 - 2.59e10i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (2.81e11 - 9.15e10i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (1.77e11 + 5.76e10i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 - 8.81e10T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-2.01e11 - 6.19e11i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
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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
−17.40015345754669372189144196689, −16.68496350509532849200617009710, −14.44570242927204798325437588282, −12.87309624254529862377961124192, −11.44374176078608358947025991196, −9.968173564353481665476459261129, −7.37086708494249988651603583710, −6.65355689128776603789124399772, −3.73290392845779402511430026534, −0.53555249909729823509657403794,
0.73748164105336904481872966021, 4.57125745220776925288916395207, 5.60821573532442667319635848007, 8.832236565869260138235109628094, 9.719045177620591441329778130861, 11.57927520699795261415235986204, 12.77674420464525969337410188967, 15.16378245528719605011840939621, 16.53380598833351015307443758809, 17.20953156149436184646783531122
