L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−1.64 + 1.50i)5-s + 1.00i·6-s + (2.19 − 2.19i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.0993 − 2.23i)10-s − 2.28i·11-s + (−0.707 − 0.707i)12-s + (−0.645 + 0.645i)13-s + 3.10i·14-s + (−0.0993 + 2.23i)15-s − 1.00·16-s + (−1.02 − 1.02i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.737 + 0.675i)5-s + 0.408i·6-s + (0.830 − 0.830i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.0314 − 0.706i)10-s − 0.689i·11-s + (−0.204 − 0.204i)12-s + (−0.179 + 0.179i)13-s + 0.830i·14-s + (−0.0256 + 0.576i)15-s − 0.250·16-s + (−0.249 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.915561 - 0.584371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.915561 - 0.584371i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.64 - 1.50i)T \) |
| 31 | \( 1 + (-5.39 + 1.36i)T \) |
good | 7 | \( 1 + (-2.19 + 2.19i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.28iT - 11T^{2} \) |
| 13 | \( 1 + (0.645 - 0.645i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.02 + 1.02i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.57iT - 19T^{2} \) |
| 23 | \( 1 + (0.646 - 0.646i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 37 | \( 1 + (5.23 + 5.23i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + (-7.37 + 7.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.53 - 2.53i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.61 + 8.61i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.51iT - 59T^{2} \) |
| 61 | \( 1 + 7.53iT - 61T^{2} \) |
| 67 | \( 1 + (0.158 - 0.158i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.65T + 71T^{2} \) |
| 73 | \( 1 + (-1.99 + 1.99i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-7.17 + 7.17i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + (4.14 - 4.14i)T - 97iT^{2} \) |
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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
−9.836760063548350386331819811177, −8.819064881388230197002159775230, −8.111977319610523132472945986208, −7.36296564773109796375274092312, −6.94603591486323592134321042616, −5.78733335429422606200603891971, −4.51803033800707790263479735068, −3.56333924161150695278819222237, −2.19538631411182010495630952911, −0.59977564617364934747376630214,
1.51402350151939471810813842579, 2.64970059565476722451491829705, 3.95628888697997632561783466461, 4.68807580580473162199016576349, 5.65775800441483020392914278633, 7.24066624354976490845831975676, 8.035994030386385727581677722375, 8.579984956252693663998454314614, 9.292773552224769273491802459785, 10.12263337133622647400407022007