| L(s) = 1 | − 3-s − 3.82·5-s − 3.82·7-s − 2·9-s + 4.82·11-s + 13-s + 3.82·15-s + 6.65·17-s − 4·19-s + 3.82·21-s + 3.17·23-s + 9.65·25-s + 5·27-s − 3.17·29-s − 4.82·33-s + 14.6·35-s − 3.82·37-s − 39-s + 2.82·41-s − 3·43-s + 7.65·45-s + 11.4·47-s + 7.65·49-s − 6.65·51-s − 3.17·53-s − 18.4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.71·5-s − 1.44·7-s − 0.666·9-s + 1.45·11-s + 0.277·13-s + 0.988·15-s + 1.61·17-s − 0.917·19-s + 0.835·21-s + 0.661·23-s + 1.93·25-s + 0.962·27-s − 0.588·29-s − 0.840·33-s + 2.47·35-s − 0.629·37-s − 0.160·39-s + 0.441·41-s − 0.457·43-s + 1.14·45-s + 1.67·47-s + 1.09·49-s − 0.932·51-s − 0.435·53-s − 2.49·55-s + 0.529·57-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 3T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.253015343179750951591283580025, −7.43002577708871257783988332091, −6.68744252414480750951189595635, −6.17696813675356189217766874794, −5.25740146196813861050792486180, −4.09677600755433771275645233741, −3.60508235626891630195647273566, −2.95564708280864717832401233156, −1.00843164166974151095249506504, 0,
1.00843164166974151095249506504, 2.95564708280864717832401233156, 3.60508235626891630195647273566, 4.09677600755433771275645233741, 5.25740146196813861050792486180, 6.17696813675356189217766874794, 6.68744252414480750951189595635, 7.43002577708871257783988332091, 8.253015343179750951591283580025
