L(s) = 1 | + i·2-s + (−1.02 − 1.39i)3-s − 4-s + i·5-s + (1.39 − 1.02i)6-s + 0.608·7-s − i·8-s + (−0.901 + 2.86i)9-s − 10-s − 1.47·11-s + (1.02 + 1.39i)12-s − 4.49i·13-s + 0.608i·14-s + (1.39 − 1.02i)15-s + 16-s − 0.826·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.591 − 0.806i)3-s − 0.5·4-s + 0.447i·5-s + (0.570 − 0.418i)6-s + 0.230·7-s − 0.353i·8-s + (−0.300 + 0.953i)9-s − 0.316·10-s − 0.444·11-s + (0.295 + 0.403i)12-s − 1.24i·13-s + 0.162i·14-s + (0.360 − 0.264i)15-s + 0.250·16-s − 0.200·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.434427 - 0.457745i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.434427 - 0.457745i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.02 + 1.39i)T \) |
| 5 | \( 1 - iT \) |
| 31 | \( 1 + (4.31 + 3.52i)T \) |
good | 7 | \( 1 - 0.608T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 4.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.826T + 17T^{2} \) |
| 19 | \( 1 + 0.806T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 37 | \( 1 - 2.79iT - 37T^{2} \) |
| 41 | \( 1 + 7.02iT - 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 4.11iT - 47T^{2} \) |
| 53 | \( 1 + 7.69T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 7.71iT - 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 + 4.02iT - 71T^{2} \) |
| 73 | \( 1 - 4.96iT - 73T^{2} \) |
| 79 | \( 1 + 4.39iT - 79T^{2} \) |
| 83 | \( 1 + 2.04T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−9.956542234256588712061332050287, −8.693749358817117229938261424268, −7.903113183131691236694784030988, −7.31524770410699679489339003186, −6.46809747579847810360856901741, −5.59577994836904961490054912034, −4.97885753971770570114332876552, −3.46758901991819306447576968692, −2.09502753681610111135441342168, −0.32712134098305778662505573679,
1.47435880814629783597474408557, 2.98476526110957139371033425346, 4.21403278610353157882805341045, 4.73679826245669841147553306149, 5.70303430466604414999832381033, 6.72073548327289236248374946189, 8.001452023503680625247849866301, 9.024567541873100727386068297946, 9.467333682752620103893415337042, 10.35697846699817492512623910393