| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−2.80 − 1.30i)3-s + (0.766 + 0.642i)4-s + (0.849 − 2.06i)5-s + (2.19 + 2.19i)6-s + (2.78 − 1.94i)7-s + (−0.500 − 0.866i)8-s + (4.23 + 5.05i)9-s + (−1.50 + 1.65i)10-s + (2.76 − 1.59i)11-s + (−1.30 − 2.80i)12-s + (3.73 + 3.13i)13-s + (−3.28 + 0.878i)14-s + (−5.09 + 4.69i)15-s + (0.173 + 0.984i)16-s + (−3.57 − 4.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−1.62 − 0.755i)3-s + (0.383 + 0.321i)4-s + (0.379 − 0.925i)5-s + (0.894 + 0.894i)6-s + (1.05 − 0.736i)7-s + (−0.176 − 0.306i)8-s + (1.41 + 1.68i)9-s + (−0.476 + 0.522i)10-s + (0.833 − 0.481i)11-s + (−0.377 − 0.810i)12-s + (1.03 + 0.869i)13-s + (−0.876 + 0.234i)14-s + (−1.31 + 1.21i)15-s + (0.0434 + 0.246i)16-s + (−0.866 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.314426 - 0.644356i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.314426 - 0.644356i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.849 + 2.06i)T \) |
| 37 | \( 1 + (4.17 - 4.42i)T \) |
| good | 3 | \( 1 + (2.80 + 1.30i)T + (1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-2.78 + 1.94i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (-2.76 + 1.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.73 - 3.13i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.57 + 4.25i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.735 + 1.57i)T + (-12.2 - 14.5i)T^{2} \) |
| 23 | \( 1 + (-3.62 + 6.28i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.49 - 9.30i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-1.89 + 1.89i)T - 31iT^{2} \) |
| 41 | \( 1 + (-3.28 + 3.90i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 + (5.01 - 1.34i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.75 - 1.92i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (5.41 + 3.79i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-6.12 + 0.535i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-4.92 - 7.03i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (2.08 - 0.760i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.18 - 7.18i)T + 73iT^{2} \) |
| 79 | \( 1 + (4.48 + 6.40i)T + (-27.0 + 74.2i)T^{2} \) |
| 83 | \( 1 + (0.702 - 8.02i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-9.05 + 12.9i)T + (-30.4 - 83.6i)T^{2} \) |
| 97 | \( 1 + (2.25 + 1.30i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.30525957563274475792275542669, −10.52811033929292078775005615990, −9.128386463550218823318195765148, −8.386183166233872494888373215318, −7.02475257809817504429364967255, −6.52243856160260704692450897907, −5.19976881149906853762332303791, −4.38659627043959061336112155390, −1.63418004475800658930502082892, −0.849125400511920496936999094654,
1.63801311883189672337043125394, 3.83747290455598037682196774724, 5.24616335706307742444681936567, 5.98282522332291303043509115083, 6.66695850304950451621664685050, 7.989627889301988373944102827071, 9.223787527312176702563272149880, 10.06986725998329751861430466943, 10.94040049801280947672910324700, 11.30469743392466261384319331381
