| L(s) = 1 | + 3.77·7-s + 4.77·11-s − 3·13-s + 6.77·17-s + 5.77·19-s + 0.772·23-s − 4.77·29-s + 10.7·31-s − 7.77·37-s + 7.54·41-s − 6.77·43-s + 2.77·47-s + 7.22·49-s − 7.54·53-s + 12·59-s − 7.77·61-s − 6.22·67-s − 3.54·71-s + 3.77·73-s + 18·77-s − 5·79-s − 6·83-s + 8·89-s − 11.3·91-s − 7.31·97-s − 18.7·101-s − 8.22·103-s + ⋯ |
| L(s) = 1 | + 1.42·7-s + 1.43·11-s − 0.832·13-s + 1.64·17-s + 1.32·19-s + 0.160·23-s − 0.886·29-s + 1.93·31-s − 1.27·37-s + 1.17·41-s − 1.03·43-s + 0.404·47-s + 1.03·49-s − 1.03·53-s + 1.56·59-s − 0.995·61-s − 0.760·67-s − 0.420·71-s + 0.441·73-s + 2.05·77-s − 0.562·79-s − 0.658·83-s + 0.847·89-s − 1.18·91-s − 0.742·97-s − 1.86·101-s − 0.810·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.941856739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941856739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 - 0.772T + 23T^{2} \) |
| 29 | \( 1 + 4.77T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 - 7.54T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 - 2.77T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + 6.22T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−8.066516376217008478471418202760, −7.53375084759905049343977821414, −6.91054563513188278563046099110, −5.89341351961823396716179942980, −5.21599824420733875388994123299, −4.60412340907756709015656996571, −3.72857498901931462702315367468, −2.86891955003130331191113877015, −1.62881691559700265198924115775, −1.05718477139330004896954336937,
1.05718477139330004896954336937, 1.62881691559700265198924115775, 2.86891955003130331191113877015, 3.72857498901931462702315367468, 4.60412340907756709015656996571, 5.21599824420733875388994123299, 5.89341351961823396716179942980, 6.91054563513188278563046099110, 7.53375084759905049343977821414, 8.066516376217008478471418202760
