| L(s) = 1 | − 12·5-s + 7·7-s + 60·11-s + 79·13-s − 108·17-s + 11·19-s + 132·23-s + 19·25-s − 96·29-s − 20·31-s − 84·35-s + 169·37-s + 192·41-s + 488·43-s − 204·47-s − 294·49-s − 360·53-s − 720·55-s + 156·59-s − 83·61-s − 948·65-s + 47·67-s − 216·71-s − 511·73-s + 420·77-s + 529·79-s − 1.12e3·83-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 0.377·7-s + 1.64·11-s + 1.68·13-s − 1.54·17-s + 0.132·19-s + 1.19·23-s + 0.151·25-s − 0.614·29-s − 0.115·31-s − 0.405·35-s + 0.750·37-s + 0.731·41-s + 1.73·43-s − 0.633·47-s − 6/7·49-s − 0.933·53-s − 1.76·55-s + 0.344·59-s − 0.174·61-s − 1.80·65-s + 0.0857·67-s − 0.361·71-s − 0.819·73-s + 0.621·77-s + 0.753·79-s − 1.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.154846148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154846148\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 79 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 11 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 169 T + p^{3} T^{2} \) |
| 41 | \( 1 - 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 488 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 - 156 T + p^{3} T^{2} \) |
| 61 | \( 1 + 83 T + p^{3} T^{2} \) |
| 67 | \( 1 - 47 T + p^{3} T^{2} \) |
| 71 | \( 1 + 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 529 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1128 T + p^{3} T^{2} \) |
| 89 | \( 1 - 36 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−8.916382037593908358155353522185, −8.281659294479560448777940817652, −7.36217468900736146820289505287, −6.59580323036070741528761322001, −5.89939166704960516338269909101, −4.47539407465553522343267804071, −4.06887708066217195287411314070, −3.19747750143013138833704672471, −1.68501874058193482137252701642, −0.74026858659953223289695311364,
0.74026858659953223289695311364, 1.68501874058193482137252701642, 3.19747750143013138833704672471, 4.06887708066217195287411314070, 4.47539407465553522343267804071, 5.89939166704960516338269909101, 6.59580323036070741528761322001, 7.36217468900736146820289505287, 8.281659294479560448777940817652, 8.916382037593908358155353522185
