L(s) = 1 | − 5-s + 2.84i·7-s − 2.95·11-s − 0.451·13-s + 4.38·17-s − 7.84i·19-s + (3.10 + 3.65i)23-s + 25-s + 3.01i·29-s − 6.09·31-s − 2.84i·35-s − 0.619i·37-s − 5.35i·41-s + 3.75i·43-s + 5.23i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.07i·7-s − 0.891·11-s − 0.125·13-s + 1.06·17-s − 1.80i·19-s + (0.647 + 0.762i)23-s + 0.200·25-s + 0.560i·29-s − 1.09·31-s − 0.481i·35-s − 0.101i·37-s − 0.836i·41-s + 0.572i·43-s + 0.763i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0887 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.276209431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276209431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-3.10 - 3.65i)T \) |
good | 7 | \( 1 - 2.84iT - 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 + 0.451T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 7.84iT - 19T^{2} \) |
| 29 | \( 1 - 3.01iT - 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 0.619iT - 37T^{2} \) |
| 41 | \( 1 + 5.35iT - 41T^{2} \) |
| 43 | \( 1 - 3.75iT - 43T^{2} \) |
| 47 | \( 1 - 5.23iT - 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 11.8iT - 59T^{2} \) |
| 61 | \( 1 - 9.81iT - 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 8.71iT - 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 + 3.64T + 89T^{2} \) |
| 97 | \( 1 - 8.46iT - 97T^{2} \) |
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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
−7.909155653289915160154184072473, −7.34129016978650373726845660698, −6.76907194302992188031854428631, −5.57150870045104864175744606791, −5.40902982809934627267988774189, −4.61750883030199761970000008248, −3.55066300840965922814637877244, −2.85786185624484312052946918044, −2.20310786823272703208662816196, −0.895570293787353494986069769893,
0.37304532714953570124214991847, 1.36686570914359103901334244816, 2.47958464088409696827767342096, 3.51119956483186872410287252511, 3.89715098959799497826825203335, 4.81419564638070618652969153875, 5.52151880804564893420129250235, 6.21694099725815739835632897033, 7.26332163153325562735160602613, 7.49476482794833337003958810441