| L(s) = 1 | + (−0.159 − 1.00i)2-s + (1.15 − 0.121i)3-s + (0.908 − 0.295i)4-s + (0.515 − 1.92i)5-s + (−0.306 − 1.14i)6-s + (0.690 − 0.448i)7-s + (−1.37 − 2.69i)8-s + (−1.62 + 0.344i)9-s + (−2.02 − 0.212i)10-s + (−2.18 + 0.114i)11-s + (1.01 − 0.450i)12-s + (−1.66 − 3.20i)13-s + (−0.563 − 0.625i)14-s + (0.360 − 2.27i)15-s + (−0.950 + 0.690i)16-s + (3.76 + 4.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.113 − 0.713i)2-s + (0.665 − 0.0699i)3-s + (0.454 − 0.147i)4-s + (0.230 − 0.859i)5-s + (−0.125 − 0.467i)6-s + (0.261 − 0.169i)7-s + (−0.484 − 0.951i)8-s + (−0.540 + 0.114i)9-s + (−0.639 − 0.0672i)10-s + (−0.657 + 0.0344i)11-s + (0.292 − 0.130i)12-s + (−0.460 − 0.887i)13-s + (−0.150 − 0.167i)14-s + (0.0932 − 0.588i)15-s + (−0.237 + 0.172i)16-s + (0.912 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05042 - 1.43987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05042 - 1.43987i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.66 + 3.20i)T \) |
| 31 | \( 1 + (-5.52 + 0.724i)T \) |
| good | 2 | \( 1 + (0.159 + 1.00i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-1.15 + 0.121i)T + (2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.515 + 1.92i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.690 + 0.448i)T + (2.84 - 6.39i)T^{2} \) |
| 11 | \( 1 + (2.18 - 0.114i)T + (10.9 - 1.14i)T^{2} \) |
| 17 | \( 1 + (-3.76 - 4.17i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.54 - 1.36i)T + (14.1 + 12.7i)T^{2} \) |
| 23 | \( 1 + (1.86 - 5.74i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.20 + 3.03i)T + (-8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-7.12 + 1.90i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (3.54 - 2.86i)T + (8.52 - 40.1i)T^{2} \) |
| 43 | \( 1 + (-4.06 + 1.80i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (0.328 - 2.07i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.53 + 7.23i)T + (-48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (4.20 + 3.40i)T + (12.2 + 57.7i)T^{2} \) |
| 61 | \( 1 + 1.27iT - 61T^{2} \) |
| 67 | \( 1 + (0.468 - 1.75i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (2.99 + 1.94i)T + (28.8 + 64.8i)T^{2} \) |
| 73 | \( 1 + (-13.2 + 0.694i)T + (72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (5.72 - 5.15i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-4.95 - 6.11i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (2.27 - 4.46i)T + (-52.3 - 72.0i)T^{2} \) |
| 97 | \( 1 + (1.04 - 2.05i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.97422287201394457807542857091, −10.00632204825148121816362125517, −9.437907082529921700914900729228, −8.088582313690688721231473721295, −7.74043664491217653106410291213, −6.01849715956734161614221937674, −5.18866162786217572151405223228, −3.54699824624631961616638405671, −2.55963816106982958253445767326, −1.21746245840541468576144406380,
2.50359964499799424438244970667, 3.00124307957674515309240551372, 4.91669888085737584022506654659, 6.05641838642472417694625882571, 6.96420100037600516456491661836, 7.75292428073222386990056759976, 8.570654073244057545175789563530, 9.569581776720269101009209474723, 10.59143912221213716735297127562, 11.58525111724587984119893592790
