| L(s) = 1 | − 1.86·3-s + 1.48·5-s − 2.55·7-s + 0.484·9-s − 1.59·11-s + 13-s − 2.77·15-s + 4.76·17-s − 3.51·19-s + 4.76·21-s − 3.73·23-s − 2.79·25-s + 4.69·27-s + 2·29-s + 0.685·31-s + 2.96·33-s − 3.79·35-s − 1.73·37-s − 1.86·39-s − 7.52·41-s + 4.69·43-s + 0.719·45-s − 5.38·47-s − 0.484·49-s − 8.89·51-s − 5.52·53-s − 2.36·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 0.664·5-s − 0.964·7-s + 0.161·9-s − 0.479·11-s + 0.277·13-s − 0.715·15-s + 1.15·17-s − 0.806·19-s + 1.03·21-s − 0.778·23-s − 0.559·25-s + 0.903·27-s + 0.371·29-s + 0.123·31-s + 0.516·33-s − 0.640·35-s − 0.285·37-s − 0.298·39-s − 1.17·41-s + 0.716·43-s + 0.107·45-s − 0.784·47-s − 0.0692·49-s − 1.24·51-s − 0.759·53-s − 0.318·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8840144774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8840144774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 0.685T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.96T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 + 4.06T + 83T^{2} \) |
| 89 | \( 1 + 0.969T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.579083276413474901045279040959, −7.924759357586246567677720392131, −6.74750089220539763522223226505, −6.35273703433689147871507650405, −5.61866150575042860226318919455, −5.15305011264213691072931863480, −3.95940743365128932279339145843, −3.05190695585005502433611463441, −1.96470195727427781537973493651, −0.57397808760970688918900023380,
0.57397808760970688918900023380, 1.96470195727427781537973493651, 3.05190695585005502433611463441, 3.95940743365128932279339145843, 5.15305011264213691072931863480, 5.61866150575042860226318919455, 6.35273703433689147871507650405, 6.74750089220539763522223226505, 7.924759357586246567677720392131, 8.579083276413474901045279040959
