L(s) = 1 | + (−2.39 + 1.38i)2-s + (−0.969 + 1.43i)3-s + (2.81 − 4.88i)4-s + (0.866 + 0.5i)5-s + (0.337 − 4.77i)6-s + (−0.814 + 0.470i)7-s + 10.0i·8-s + (−1.11 − 2.78i)9-s − 2.76·10-s + (1.21 − 0.701i)11-s + (4.27 + 8.77i)12-s + (−0.345 − 3.58i)13-s + (1.29 − 2.25i)14-s + (−1.55 + 0.758i)15-s + (−8.25 − 14.2i)16-s + 4.46·17-s + ⋯ |
L(s) = 1 | + (−1.69 + 0.977i)2-s + (−0.559 + 0.828i)3-s + (1.40 − 2.44i)4-s + (0.387 + 0.223i)5-s + (0.137 − 1.94i)6-s + (−0.307 + 0.177i)7-s + 3.55i·8-s + (−0.373 − 0.927i)9-s − 0.873·10-s + (0.366 − 0.211i)11-s + (1.23 + 2.53i)12-s + (−0.0957 − 0.995i)13-s + (0.347 − 0.601i)14-s + (−0.402 + 0.195i)15-s + (−2.06 − 3.57i)16-s + 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0925119 + 0.463208i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0925119 + 0.463208i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.969 - 1.43i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.345 + 3.58i)T \) |
good | 2 | \( 1 + (2.39 - 1.38i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (0.814 - 0.470i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.701i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 4.15iT - 19T^{2} \) |
| 23 | \( 1 + (3.91 - 6.78i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.28 - 7.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.20 - 3.58i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.11iT - 37T^{2} \) |
| 41 | \( 1 + (5.13 + 2.96i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 0.635i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.03 - 1.75i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + (1.86 + 1.07i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 6.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.82 - 5.67i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.61iT - 71T^{2} \) |
| 73 | \( 1 + 2.38iT - 73T^{2} \) |
| 79 | \( 1 + (3.36 + 5.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.1 - 7.56i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.36iT - 89T^{2} \) |
| 97 | \( 1 + (2.13 - 1.23i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.45517578491812971347963452196, −10.10354558582734052297717440520, −9.448172003878725837761935175925, −8.534149198633635761415893008747, −7.68745519789371173501190842303, −6.59117447598992258091091561745, −5.81467751637667009665470366530, −5.26112702614128210656720200682, −3.21339946610604132478588854313, −1.24310737191129087526758227770,
0.56543546206931560650949148659, 1.79370100168172598886161792715, 2.78541853660275446952128099517, 4.43444047852105498782595624422, 6.39724026874209044969591163778, 6.81885030125591940321901698794, 8.003038818773014856131807874563, 8.525916810657801830257825508851, 9.805904132225415850349747807923, 10.01835759790113003725527428235