| L(s) = 1 | + (−1.83 + 1.13i)2-s + (−2.62 − 0.150i)3-s + (1.18 − 2.39i)4-s + (1.69 + 1.45i)5-s + (4.98 − 2.70i)6-s + (−2.79 + 1.59i)7-s + (0.137 + 1.43i)8-s + (3.86 + 0.445i)9-s + (−4.77 − 0.739i)10-s + (2.11 − 2.03i)11-s + (−3.46 + 6.08i)12-s + (5.84 − 2.22i)13-s + (3.32 − 6.11i)14-s + (−4.22 − 4.07i)15-s + (1.34 + 1.76i)16-s + (0.947 + 1.82i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 + 0.805i)2-s + (−1.51 − 0.0870i)3-s + (0.591 − 1.19i)4-s + (0.758 + 0.651i)5-s + (2.03 − 1.10i)6-s + (−1.05 + 0.601i)7-s + (0.0487 + 0.507i)8-s + (1.28 + 0.148i)9-s + (−1.50 − 0.233i)10-s + (0.637 − 0.613i)11-s + (−0.999 + 1.75i)12-s + (1.62 − 0.618i)13-s + (0.888 − 1.63i)14-s + (−1.09 − 1.05i)15-s + (0.336 + 0.441i)16-s + (0.229 + 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.137419 + 0.350673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137419 + 0.350673i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.69 - 1.45i)T \) |
| 83 | \( 1 + (2.91 + 8.63i)T \) |
| good | 2 | \( 1 + (1.83 - 1.13i)T + (0.887 - 1.79i)T^{2} \) |
| 3 | \( 1 + (2.62 + 0.150i)T + (2.98 + 0.344i)T^{2} \) |
| 7 | \( 1 + (2.79 - 1.59i)T + (3.57 - 6.01i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 2.03i)T + (0.421 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-5.84 + 2.22i)T + (9.70 - 8.64i)T^{2} \) |
| 17 | \( 1 + (-0.947 - 1.82i)T + (-9.78 + 13.9i)T^{2} \) |
| 19 | \( 1 + (-0.610 - 3.14i)T + (-17.6 + 7.10i)T^{2} \) |
| 23 | \( 1 + (3.51 + 1.65i)T + (14.6 + 17.7i)T^{2} \) |
| 29 | \( 1 + (0.540 + 1.34i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (3.55 - 2.92i)T + (5.90 - 30.4i)T^{2} \) |
| 37 | \( 1 + (-4.94 + 6.23i)T + (-8.43 - 36.0i)T^{2} \) |
| 41 | \( 1 + (8.84 - 2.06i)T + (36.7 - 18.1i)T^{2} \) |
| 43 | \( 1 + (-1.33 - 7.65i)T + (-40.4 + 14.5i)T^{2} \) |
| 47 | \( 1 + (-1.99 - 9.34i)T + (-42.8 + 19.2i)T^{2} \) |
| 53 | \( 1 + (1.95 - 9.13i)T + (-48.3 - 21.6i)T^{2} \) |
| 59 | \( 1 + (7.72 - 0.296i)T + (58.8 - 4.51i)T^{2} \) |
| 61 | \( 1 + (3.94 - 2.34i)T + (29.1 - 53.5i)T^{2} \) |
| 67 | \( 1 + (-7.45 - 1.00i)T + (64.6 + 17.7i)T^{2} \) |
| 71 | \( 1 + (-10.7 + 2.95i)T + (61.0 - 36.2i)T^{2} \) |
| 73 | \( 1 + (2.17 + 2.97i)T + (-22.0 + 69.5i)T^{2} \) |
| 79 | \( 1 + (-9.18 - 4.55i)T + (47.8 + 62.8i)T^{2} \) |
| 89 | \( 1 + (-1.34 - 2.47i)T + (-48.3 + 74.7i)T^{2} \) |
| 97 | \( 1 + (-3.18 - 10.7i)T + (-81.4 + 52.7i)T^{2} \) |
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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
−11.05189468002699179201550388161, −10.61464134054028462229777975082, −9.725583271417377726398099424413, −8.959300172289969285644064516553, −7.81760523487254297076314616036, −6.45712243340073429521893830740, −6.15203103038880267090246385924, −5.76928317958802517290072337767, −3.50132817739761952844159587300, −1.21938368648659309982777708712,
0.54139250328739314939774032661, 1.69571654324682635973206557037, 3.75111445275891439412589973865, 5.17723766056187502826394400211, 6.27983400747072014013062898870, 6.96808398341201277340647109670, 8.541464686599053734322725007341, 9.477722350138441441775443707726, 9.938200702565718988138995811368, 10.77108371647125523589069732556
