L(s) = 1 | + i·2-s + (1.71 − 0.206i)3-s − 4-s − i·5-s + (0.206 + 1.71i)6-s − 1.18·7-s − i·8-s + (2.91 − 0.711i)9-s + 10-s − 1.57·11-s + (−1.71 + 0.206i)12-s − 3.43i·13-s − 1.18i·14-s + (−0.206 − 1.71i)15-s + 16-s + 4.55·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.992 − 0.119i)3-s − 0.5·4-s − 0.447i·5-s + (0.0844 + 0.702i)6-s − 0.446·7-s − 0.353i·8-s + (0.971 − 0.237i)9-s + 0.316·10-s − 0.475·11-s + (−0.496 + 0.0596i)12-s − 0.953i·13-s − 0.315i·14-s + (−0.0533 − 0.444i)15-s + 0.250·16-s + 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09569 + 0.105579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09569 + 0.105579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.71 + 0.206i)T \) |
| 5 | \( 1 + iT \) |
| 31 | \( 1 + (5.56 + 0.105i)T \) |
good | 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 + 3.43iT - 13T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 37 | \( 1 + 9.82iT - 37T^{2} \) |
| 41 | \( 1 - 5.42iT - 41T^{2} \) |
| 43 | \( 1 - 6.82iT - 43T^{2} \) |
| 47 | \( 1 + 1.46iT - 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 + 6.84iT - 59T^{2} \) |
| 61 | \( 1 + 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 6.52iT - 71T^{2} \) |
| 73 | \( 1 - 4.67iT - 73T^{2} \) |
| 79 | \( 1 - 6.13iT - 79T^{2} \) |
| 83 | \( 1 - 3.86T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−9.737328738295501416180693031702, −9.242222100765257340556378164590, −8.121552313155874631143094818849, −7.79267857474204427313188217096, −6.88624198704910494745961790948, −5.69949644719968593176899402464, −4.94148236387645159430872184753, −3.62871850861648337862229737068, −2.85707349447426191020522937357, −1.04398508478023221137766804694,
1.44852276090241037486706787858, 2.80273006729021991871760009389, 3.34081067333359446811334215393, 4.43020455438681106182905854625, 5.53753785237939754238886167488, 6.91018572057334117487010750428, 7.59913637019183096893321626625, 8.583675756373418500895296984126, 9.365246380720600544915352063033, 10.04998162374218119812397537968