| L(s) = 1 | − 3-s + 0.0621·5-s + 0.782·7-s + 9-s + 2.68·11-s + 6.79·13-s − 0.0621·15-s + 4.20·17-s + 7.94·19-s − 0.782·21-s + 5.42·23-s − 4.99·25-s − 27-s + 1.95·29-s + 2.86·31-s − 2.68·33-s + 0.0486·35-s − 7.68·37-s − 6.79·39-s − 2.03·41-s − 5.17·43-s + 0.0621·45-s − 7.62·47-s − 6.38·49-s − 4.20·51-s + 3.26·53-s + 0.166·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0278·5-s + 0.295·7-s + 0.333·9-s + 0.809·11-s + 1.88·13-s − 0.0160·15-s + 1.02·17-s + 1.82·19-s − 0.170·21-s + 1.13·23-s − 0.999·25-s − 0.192·27-s + 0.363·29-s + 0.514·31-s − 0.467·33-s + 0.00822·35-s − 1.26·37-s − 1.08·39-s − 0.317·41-s − 0.788·43-s + 0.00926·45-s − 1.11·47-s − 0.912·49-s − 0.589·51-s + 0.448·53-s + 0.0225·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.182693911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.182693911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.0621T + 5T^{2} \) |
| 7 | \( 1 - 0.782T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 7.94T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + 2.03T + 41T^{2} \) |
| 43 | \( 1 + 5.17T + 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 - 2.33T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 1.25T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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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
−8.376969640475237023438422341592, −7.76320491567568261025143927837, −6.82718316178803930191483804000, −6.26001402434113549400508381137, −5.45265083413668456711620842441, −4.87595855438250489137399425021, −3.64466482481077932414300845478, −3.30768278110998567997457707315, −1.56126527380372201695552480189, −1.01052976772749059155210914057,
1.01052976772749059155210914057, 1.56126527380372201695552480189, 3.30768278110998567997457707315, 3.64466482481077932414300845478, 4.87595855438250489137399425021, 5.45265083413668456711620842441, 6.26001402434113549400508381137, 6.82718316178803930191483804000, 7.76320491567568261025143927837, 8.376969640475237023438422341592
