| L(s) = 1 | − 0.786·5-s − 4.29·7-s − 1.21·11-s + 5.08·13-s + 2.29·17-s + 19-s + 7.67·23-s − 4.38·25-s + 0.489·29-s + 3.38·35-s − 2·37-s + 4.16·41-s − 12.9·43-s − 5.80·47-s + 11.4·49-s − 1.93·53-s + 0.954·55-s − 11.0·59-s − 5.38·61-s − 4·65-s − 2.48·67-s − 11.7·71-s − 8.46·73-s + 5.21·77-s − 1.83·79-s − 7.02·83-s − 1.80·85-s + ⋯ |
| L(s) = 1 | − 0.351·5-s − 1.62·7-s − 0.365·11-s + 1.41·13-s + 0.557·17-s + 0.229·19-s + 1.60·23-s − 0.876·25-s + 0.0909·29-s + 0.571·35-s − 0.328·37-s + 0.650·41-s − 1.97·43-s − 0.847·47-s + 1.63·49-s − 0.266·53-s + 0.128·55-s − 1.44·59-s − 0.688·61-s − 0.496·65-s − 0.304·67-s − 1.39·71-s − 0.990·73-s + 0.594·77-s − 0.206·79-s − 0.770·83-s − 0.196·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.786T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 0.489T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 3.57T + 97T^{2} \) |
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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
−8.533839834445068816264161259759, −7.64972159253528987301320643714, −6.85555895998248098128701314468, −6.19420992362612097486693823114, −5.50575520742664483005463767156, −4.37163683332581426417877666573, −3.29875580263938414283667095713, −3.09097488233587200884320823064, −1.38592291591438093795665733606, 0,
1.38592291591438093795665733606, 3.09097488233587200884320823064, 3.29875580263938414283667095713, 4.37163683332581426417877666573, 5.50575520742664483005463767156, 6.19420992362612097486693823114, 6.85555895998248098128701314468, 7.64972159253528987301320643714, 8.533839834445068816264161259759
