L(s) = 1 | + (0.707 − 0.707i)2-s + (1.68 + 0.404i)3-s − 1.00i·4-s + (2.01 − 0.959i)5-s + (1.47 − 0.904i)6-s + (−2.95 − 2.95i)7-s + (−0.707 − 0.707i)8-s + (2.67 + 1.36i)9-s + (0.749 − 2.10i)10-s − 4.99i·11-s + (0.404 − 1.68i)12-s + (−0.576 + 0.576i)13-s − 4.18·14-s + (3.78 − 0.799i)15-s − 1.00·16-s + (−1.92 + 1.92i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.972 + 0.233i)3-s − 0.500i·4-s + (0.903 − 0.429i)5-s + (0.602 − 0.369i)6-s + (−1.11 − 1.11i)7-s + (−0.250 − 0.250i)8-s + (0.890 + 0.454i)9-s + (0.236 − 0.666i)10-s − 1.50i·11-s + (0.116 − 0.486i)12-s + (−0.159 + 0.159i)13-s − 1.11·14-s + (0.978 − 0.206i)15-s − 0.250·16-s + (−0.467 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02256 - 2.07158i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02256 - 2.07158i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.68 - 0.404i)T \) |
| 5 | \( 1 + (-2.01 + 0.959i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (2.95 + 2.95i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.99iT - 11T^{2} \) |
| 13 | \( 1 + (0.576 - 0.576i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.92 - 1.92i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.28iT - 19T^{2} \) |
| 23 | \( 1 + (-2.35 - 2.35i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 37 | \( 1 + (-5.67 - 5.67i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.89iT - 41T^{2} \) |
| 43 | \( 1 + (-7.51 + 7.51i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.39 + 4.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.78 - 3.78i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 + (-8.10 - 8.10i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.808iT - 71T^{2} \) |
| 73 | \( 1 + (-6.98 + 6.98i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.04iT - 79T^{2} \) |
| 83 | \( 1 + (5.46 + 5.46i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + (-9.00 - 9.00i)T + 97iT^{2} \) |
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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
−9.859227673047734346689596393833, −9.229120201095454480639046218630, −8.439757713736295982350579477031, −7.25154029712816976234868142842, −6.30082278284902142565200607690, −5.44001483530481367085691354932, −4.09970605514757414415163859719, −3.51394454798393515610654446960, −2.47776498951456318965874936922, −1.08029058902690369282257554238,
2.29007143575707735666625175575, 2.62192716102275505956110518347, 3.91446189267061604701871041770, 5.14009328728176397174127092093, 6.09690531502497224070731258813, 6.94128569979023536757834303287, 7.42152105057046761545497542820, 8.799904934945121847414303213041, 9.432670849197105614280510326144, 9.771269481507185677608196813054