L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (0.433 − 2.19i)5-s − 1.00i·6-s + (0.130 − 0.130i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−1.24 − 1.85i)10-s + 2.37i·11-s + (−0.707 − 0.707i)12-s + (0.369 − 0.369i)13-s − 0.184i·14-s + (−1.24 − 1.85i)15-s − 1.00·16-s + (−2.11 − 2.11i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (0.193 − 0.981i)5-s − 0.408i·6-s + (0.0492 − 0.0492i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.393 − 0.587i)10-s + 0.717i·11-s + (−0.204 − 0.204i)12-s + (0.102 − 0.102i)13-s − 0.0492i·14-s + (−0.321 − 0.479i)15-s − 0.250·16-s + (−0.513 − 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.896594 - 2.09589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.896594 - 2.09589i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.433 + 2.19i)T \) |
| 31 | \( 1 + (0.0759 - 5.56i)T \) |
good | 7 | \( 1 + (-0.130 + 0.130i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.37iT - 11T^{2} \) |
| 13 | \( 1 + (-0.369 + 0.369i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.11 + 2.11i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.54iT - 19T^{2} \) |
| 23 | \( 1 + (-4.19 + 4.19i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 37 | \( 1 + (-3.71 - 3.71i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 + (8.21 - 8.21i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.40 + 5.40i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.93 - 2.93i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.137iT - 59T^{2} \) |
| 61 | \( 1 + 0.854iT - 61T^{2} \) |
| 67 | \( 1 + (-4.40 + 4.40i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + (-1.62 + 1.62i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + (-8.96 + 8.96i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (-2.77 + 2.77i)T - 97iT^{2} \) |
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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
−9.652953450012411779233445091200, −8.998004394898727088247726080493, −8.274665615173914584460110950491, −7.07322286534688169199172946696, −6.36603281114145893760122395660, −4.80236400783902508842944870710, −4.77247389444301840321356989433, −3.16467678245033545189409790508, −2.14470028540360569054107902009, −0.888170791445799837504160110445,
2.10582087892606623014211431597, 3.34583041862966132217635392496, 3.89860609836066279802498864162, 5.26777689340546689031929108176, 6.06636194220720750041351015122, 6.87782417847511024525365294353, 7.85312832770787296740556266036, 8.533929446905649909957742687491, 9.559312204395428164378044356739, 10.37886914323134063928657158642