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
−10.01835759790113003725527428235, −9.805904132225415850349747807923, −8.525916810657801830257825508851, −8.003038818773014856131807874563, −6.81885030125591940321901698794, −6.39724026874209044969591163778, −4.43444047852105498782595624422, −2.78541853660275446952128099517, −1.79370100168172598886161792715, −0.56543546206931560650949148659,
1.24310737191129087526758227770, 3.21339946610604132478588854313, 5.26112702614128210656720200682, 5.81467751637667009665470366530, 6.59117447598992258091091561745, 7.68745519789371173501190842303, 8.534149198633635761415893008747, 9.448172003878725837761935175925, 10.10354558582734052297717440520, 10.45517578491812971347963452196