| L(s) = 1 | + (−1.10 − 0.878i)2-s + (0.456 + 1.94i)4-s − i·5-s + 1.38i·7-s + (1.20 − 2.55i)8-s + (−0.878 + 1.10i)10-s − 5.13·11-s + 13-s + (1.21 − 1.53i)14-s + (−3.58 + 1.77i)16-s − 3.53i·17-s − 0.325i·19-s + (1.94 − 0.456i)20-s + (5.69 + 4.50i)22-s + 0.980·23-s + ⋯ |
| L(s) = 1 | + (−0.783 − 0.621i)2-s + (0.228 + 0.973i)4-s − 0.447i·5-s + 0.522i·7-s + (0.425 − 0.904i)8-s + (−0.277 + 0.350i)10-s − 1.54·11-s + 0.277·13-s + (0.324 − 0.409i)14-s + (−0.895 + 0.444i)16-s − 0.856i·17-s − 0.0747i·19-s + (0.435 − 0.102i)20-s + (1.21 + 0.961i)22-s + 0.204·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5643313224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5643313224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 + 0.878i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 1.38iT - 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 17 | \( 1 + 3.53iT - 17T^{2} \) |
| 19 | \( 1 + 0.325iT - 19T^{2} \) |
| 23 | \( 1 - 0.980T + 23T^{2} \) |
| 29 | \( 1 + 1.62iT - 29T^{2} \) |
| 31 | \( 1 - 0.932iT - 31T^{2} \) |
| 37 | \( 1 - 2.94T + 37T^{2} \) |
| 41 | \( 1 - 4.77iT - 41T^{2} \) |
| 43 | \( 1 - 6.22iT - 43T^{2} \) |
| 47 | \( 1 + 0.275T + 47T^{2} \) |
| 53 | \( 1 - 7.49iT - 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 15.6iT - 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.04iT - 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 - 7.21iT - 89T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−9.150413657035343532288005323541, −8.453547699460489644688182141587, −7.82237414520828358547389242976, −7.14262677653576539814664754951, −5.99670826397490828940608081057, −5.11160529123628381397216831961, −4.25459220802183083141859467233, −2.95674925182688214761614168139, −2.43976647644698215993417181359, −1.08052902035220890453418295718,
0.27909477863646139936671867683, 1.77049898834915749494191012721, 2.82762950527856992822103173644, 4.05121602190833384504408032931, 5.16843012602677008611098467559, 5.81129078876726872074679772943, 6.70634821441818943911483224843, 7.40514056630305871860726649840, 8.021998712682221784045123274256, 8.662161220873920983145444931341
