L(s) = 1 | + 2-s + (−0.828 − 1.52i)3-s + 4-s + i·5-s + (−0.828 − 1.52i)6-s + 4.83·7-s + 8-s + (−1.62 + 2.52i)9-s + i·10-s + 2i·11-s + (−0.828 − 1.52i)12-s − 1.04i·13-s + 4.83·14-s + (1.52 − 0.828i)15-s + 16-s − 0.140i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.478 − 0.878i)3-s + 0.5·4-s + 0.447i·5-s + (−0.338 − 0.620i)6-s + 1.82·7-s + 0.353·8-s + (−0.542 + 0.840i)9-s + 0.316i·10-s + 0.603i·11-s + (−0.239 − 0.439i)12-s − 0.288i·13-s + 1.29·14-s + (0.392 − 0.213i)15-s + 0.250·16-s − 0.0339i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21001 - 0.466691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21001 - 0.466691i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.828 + 1.52i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-2.65 - 3.45i)T \) |
good | 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 1.04iT - 13T^{2} \) |
| 17 | \( 1 + 0.140iT - 17T^{2} \) |
| 23 | \( 1 + 6.63iT - 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 3.58iT - 31T^{2} \) |
| 37 | \( 1 + 6.36iT - 37T^{2} \) |
| 41 | \( 1 + 1.18T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 4.76iT - 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 - 0.350T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 + 10.0iT - 79T^{2} \) |
| 83 | \( 1 - 10.0iT - 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 7.67iT - 97T^{2} \) |
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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
−10.94938073169735128172872860810, −10.24701004944703738801201577232, −8.540979316603466810279749919367, −7.72625904813048555210817072646, −7.14824913347893468866812687173, −5.97321948829201935432346440706, −5.17623454458643915338880145215, −4.25752429350191604217897996941, −2.51734000833866675669940784817, −1.51477449168607220591880520593,
1.48543300797896228836544286738, 3.28907727686202117499889749047, 4.46373620669476606893037496390, 5.08520936425514045040351055959, 5.74085337634174894336338968406, 7.12326098786524890458460461463, 8.233428333138003747627912220837, 8.999743545250762201362075047290, 10.13241922869102860839000278490, 11.15648165677127295917132329510