L(s) = 1 | + i·2-s + i·3-s − 4-s + 0.266i·5-s − 6-s + (−1.34 + 2.27i)7-s − i·8-s − 9-s − 0.266·10-s + (1.62 + 2.89i)11-s − i·12-s + 2.55·13-s + (−2.27 − 1.34i)14-s − 0.266·15-s + 16-s − 4.96·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.118i·5-s − 0.408·6-s + (−0.509 + 0.860i)7-s − 0.353i·8-s − 0.333·9-s − 0.0841·10-s + (0.489 + 0.871i)11-s − 0.288i·12-s + 0.707·13-s + (−0.608 − 0.360i)14-s − 0.0687·15-s + 0.250·16-s − 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0109078 + 0.945442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0109078 + 0.945442i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 + (1.34 - 2.27i)T \) |
| 11 | \( 1 + (-1.62 - 2.89i)T \) |
good | 5 | \( 1 - 0.266iT - 5T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 5.25iT - 29T^{2} \) |
| 31 | \( 1 + 4.28iT - 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 6.16iT - 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 1.78iT - 89T^{2} \) |
| 97 | \( 1 - 0.146iT - 97T^{2} \) |
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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
−11.35012878551275345295838722398, −10.52217361244659156082654698209, −9.396834838991963182931112496688, −8.930083349417372615985877403747, −7.976394331394622204515165713350, −6.50290080043924411874899485838, −6.19290728929609940828275838191, −4.75488479880105888650542278431, −3.95816995592289681949561694689, −2.36227100832675833213692549851,
0.57465128525466368950651141976, 2.14971967096039729419680063089, 3.59594560397283650666252007269, 4.44911936742441253453357691234, 6.15693433913980228134402679183, 6.67338078631146059544713654704, 8.150777483016646669478771717714, 8.738136075240206731454073330129, 9.819242276708050768565603417455, 10.94833157624843670117650305768