L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.0980 + 1.72i)3-s − 1.00i·4-s + (−0.278 + 2.21i)5-s + (−1.29 − 1.15i)6-s + (−1.59 − 1.59i)7-s + (0.707 + 0.707i)8-s + (−2.98 + 0.339i)9-s + (−1.37 − 1.76i)10-s + 3.70i·11-s + (1.72 − 0.0980i)12-s + (−1.02 + 1.02i)13-s + 2.25·14-s + (−3.86 − 0.264i)15-s − 1.00·16-s + (0.479 − 0.479i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.0566 + 0.998i)3-s − 0.500i·4-s + (−0.124 + 0.992i)5-s + (−0.527 − 0.470i)6-s + (−0.602 − 0.602i)7-s + (0.250 + 0.250i)8-s + (−0.993 + 0.113i)9-s + (−0.433 − 0.558i)10-s + 1.11i·11-s + (0.499 − 0.0283i)12-s + (−0.284 + 0.284i)13-s + 0.602·14-s + (−0.997 − 0.0682i)15-s − 0.250·16-s + (0.116 − 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.200164 - 0.271484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.200164 - 0.271484i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.0980 - 1.72i)T \) |
| 5 | \( 1 + (0.278 - 2.21i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (1.59 + 1.59i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.70iT - 11T^{2} \) |
| 13 | \( 1 + (1.02 - 1.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.479 + 0.479i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.296iT - 19T^{2} \) |
| 23 | \( 1 + (1.49 + 1.49i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 37 | \( 1 + (2.24 + 2.24i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.01iT - 41T^{2} \) |
| 43 | \( 1 + (7.37 - 7.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.230 - 0.230i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.282 + 0.282i)T + 53iT^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + (9.42 + 9.42i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.43iT - 71T^{2} \) |
| 73 | \( 1 + (5.91 - 5.91i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.56iT - 79T^{2} \) |
| 83 | \( 1 + (1.95 + 1.95i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + (-5.06 - 5.06i)T + 97iT^{2} \) |
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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
−10.18900009138952484133149357125, −10.02848644614192173729893250355, −9.196465458646355154525366294034, −8.119250409315158598555781954089, −7.16960750179731589857580943092, −6.63060210544324341439634685276, −5.51189641906409409362509490713, −4.41794752869720112777922015709, −3.54169046748048631187663183164, −2.30647945238761732615322848890,
0.18444978821250284816287838923, 1.45225022268877060210336903786, 2.70933795505514056167974851940, 3.69890052251244771574386974244, 5.29822606296862626167790814900, 5.99923129386236256741803430434, 7.07956864021863832606506051967, 8.128868944911123185795272513491, 8.564547628616181447681122939989, 9.292409345585814801633061148637