| L(s) = 1 | − 2.30i·3-s − i·5-s + 1.30·7-s − 2.30·9-s + 2.73i·11-s − 5.36i·13-s − 2.30·15-s − 1.04·17-s + 3.46i·19-s − 3i·21-s + 23-s − 25-s − 1.60i·27-s − 7.43i·29-s + 4.92·31-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s − 0.447i·5-s + 0.492·7-s − 0.767·9-s + 0.823i·11-s − 1.48i·13-s − 0.594·15-s − 0.253·17-s + 0.793i·19-s − 0.654i·21-s + 0.208·23-s − 0.200·25-s − 0.308i·27-s − 1.38i·29-s + 0.884·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435685791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435685791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30iT - 3T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 2.73iT - 11T^{2} \) |
| 13 | \( 1 + 5.36iT - 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 29 | \( 1 + 7.43iT - 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 - 1.43iT - 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 + 1.90iT - 43T^{2} \) |
| 47 | \( 1 + 5.87T + 47T^{2} \) |
| 53 | \( 1 + 7.94iT - 53T^{2} \) |
| 59 | \( 1 + 3.68iT - 59T^{2} \) |
| 61 | \( 1 + 8.81iT - 61T^{2} \) |
| 67 | \( 1 + 3.68iT - 67T^{2} \) |
| 71 | \( 1 - 1.04T + 71T^{2} \) |
| 73 | \( 1 + 0.697T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 - 5.14iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 9.46T + 97T^{2} \) |
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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
−8.203325416368524699732995369836, −7.53944636633957888516106216021, −6.74496386708861110182576409416, −6.04475798693803621986228814078, −5.21087226051172037626372499415, −4.46881246742655505032462799422, −3.28533760639454809351624614644, −2.20073833912828329593275312550, −1.48985763655220952577387101361, −0.42035974996801168205998933704,
1.49146935456393967674080967538, 2.78237877330080960653295101921, 3.52847980696286178069373916955, 4.45249188575100250331918143068, 4.83686631548038280482834113321, 5.81318765182037609838561744439, 6.70155034499921610507635202502, 7.31003373148967198753305870828, 8.579596241157168416379819768801, 8.822748477385109209771908882350
