| L(s) = 1 | − 3.16·5-s + 1.16·7-s + 4·13-s − 0.837·17-s − 5.16·19-s − 23-s + 5.00·25-s + 4.32·29-s − 6.32·31-s − 3.67·35-s + 4.32·37-s + 2·41-s + 5.16·43-s − 12.3·47-s − 5.64·49-s + 13.4·53-s + 4.32·59-s − 12.3·61-s − 12.6·65-s + 5.16·67-s + 10.6·73-s − 6.83·79-s − 4·83-s + 2.64·85-s − 9.48·89-s + 4.64·91-s + 16.3·95-s + ⋯ |
| L(s) = 1 | − 1.41·5-s + 0.439·7-s + 1.10·13-s − 0.203·17-s − 1.18·19-s − 0.208·23-s + 1.00·25-s + 0.803·29-s − 1.13·31-s − 0.621·35-s + 0.710·37-s + 0.312·41-s + 0.787·43-s − 1.79·47-s − 0.807·49-s + 1.85·53-s + 0.563·59-s − 1.57·61-s − 1.56·65-s + 0.630·67-s + 1.24·73-s − 0.769·79-s − 0.439·83-s + 0.287·85-s − 1.00·89-s + 0.487·91-s + 1.67·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 0.837T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 29 | \( 1 - 4.32T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.312835467365175670728191885099, −7.66522609851398853369959075902, −6.82280102753825951190763319164, −6.10446187748712085517077055499, −5.06463156481600982890158561580, −4.14520675968491153600711531771, −3.79078231550493493346306102836, −2.63926694156871457873624545272, −1.35046955286867310480595970232, 0,
1.35046955286867310480595970232, 2.63926694156871457873624545272, 3.79078231550493493346306102836, 4.14520675968491153600711531771, 5.06463156481600982890158561580, 6.10446187748712085517077055499, 6.82280102753825951190763319164, 7.66522609851398853369959075902, 8.312835467365175670728191885099
