| 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
| \(L(2)\) |
\(\approx\) |
\(0.5858183057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5858183057\) |
| \(L(\frac{5}{2})\) |
|
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
−8.731720408949213723544000450016, −8.145160784999690714632140150138, −7.58252513759878242579728998127, −6.50742360848819572866812198993, −5.84329010135395616699881105959, −4.68486594615514143351772935060, −3.92902658969350799008952303564, −3.09049392685859135768109995391, −1.94675074622243008283521464427, −0.34560588440245853907039467773,
0.34560588440245853907039467773, 1.94675074622243008283521464427, 3.09049392685859135768109995391, 3.92902658969350799008952303564, 4.68486594615514143351772935060, 5.84329010135395616699881105959, 6.50742360848819572866812198993, 7.58252513759878242579728998127, 8.145160784999690714632140150138, 8.731720408949213723544000450016
