| L(s) = 1 | − 2.35·3-s − 2.23i·5-s + 5.28i·7-s − 3.45·9-s + 4.16i·11-s − 19.7·13-s + 5.26i·15-s − 3.83i·17-s − 8.80i·19-s − 12.4i·21-s + (1.35 − 22.9i)23-s − 5.00·25-s + 29.3·27-s − 43.8·29-s + 3.73·31-s + ⋯ |
| L(s) = 1 | − 0.784·3-s − 0.447i·5-s + 0.755i·7-s − 0.383·9-s + 0.378i·11-s − 1.51·13-s + 0.351i·15-s − 0.225i·17-s − 0.463i·19-s − 0.592i·21-s + (0.0589 − 0.998i)23-s − 0.200·25-s + 1.08·27-s − 1.51·29-s + 0.120·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8175366094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8175366094\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-1.35 + 22.9i)T \) |
| good | 3 | \( 1 + 2.35T + 9T^{2} \) |
| 7 | \( 1 - 5.28iT - 49T^{2} \) |
| 11 | \( 1 - 4.16iT - 121T^{2} \) |
| 13 | \( 1 + 19.7T + 169T^{2} \) |
| 17 | \( 1 + 3.83iT - 289T^{2} \) |
| 19 | \( 1 + 8.80iT - 361T^{2} \) |
| 29 | \( 1 + 43.8T + 841T^{2} \) |
| 31 | \( 1 - 3.73T + 961T^{2} \) |
| 37 | \( 1 - 39.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.34T + 1.68e3T^{2} \) |
| 43 | \( 1 + 0.906iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 17.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 14.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 72.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 3.88T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 60.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 128. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100. iT - 9.40e3T^{2} \) |
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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
−9.077746176999810500373661790134, −8.374874431230785737511348406551, −7.38888429535804978032282811802, −6.62373834019605392434311099501, −5.66713624660784708752606090822, −5.10156751041771662257932392554, −4.42411018250446121629889580680, −2.92338678404166929715099357505, −2.07444502416087631441953807808, −0.47745496874223399721871025145,
0.50210272889475094715325937481, 2.00944674156568628332216078343, 3.20299536409602953364237566073, 4.10916955539595457717153944339, 5.21966886792503969370922716666, 5.75074529708748427795994485004, 6.73843481282121681405675032150, 7.40638843670388128108007652917, 8.074608290994723328942706302651, 9.295182296914738469615733067617
