| L(s) = 1 | + 3.12·3-s − 2.42·5-s − 2.20·7-s + 6.76·9-s − 13-s − 7.57·15-s − 3.33·17-s − 0.955·19-s − 6.89·21-s + 2.75·23-s + 0.879·25-s + 11.7·27-s + 8.74·29-s + 5.36·31-s + 5.35·35-s + 5.42·37-s − 3.12·39-s − 6.68·41-s + 7.04·43-s − 16.4·45-s + 11.3·47-s − 2.12·49-s − 10.4·51-s + 5.15·53-s − 2.98·57-s − 12.8·59-s − 0.162·61-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 1.08·5-s − 0.834·7-s + 2.25·9-s − 0.277·13-s − 1.95·15-s − 0.808·17-s − 0.219·19-s − 1.50·21-s + 0.574·23-s + 0.175·25-s + 2.26·27-s + 1.62·29-s + 0.962·31-s + 0.904·35-s + 0.892·37-s − 0.500·39-s − 1.04·41-s + 1.07·43-s − 2.44·45-s + 1.65·47-s − 0.304·49-s − 1.45·51-s + 0.708·53-s − 0.395·57-s − 1.67·59-s − 0.0207·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.849495985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.849495985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 0.955T + 19T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 0.162T + 61T^{2} \) |
| 67 | \( 1 + 5.50T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 0.617T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 0.388T + 97T^{2} \) |
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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
−8.073364372012518504389811101694, −7.52447671513653828914458716255, −6.88463793489564166246625751736, −6.21562287979366289458103843963, −4.69880466218716137446323913348, −4.25996934684252686182441763976, −3.45788261708681831956450731070, −2.88181535876734198107032435731, −2.20986511224140727355685852620, −0.791457524114377167129080596682,
0.791457524114377167129080596682, 2.20986511224140727355685852620, 2.88181535876734198107032435731, 3.45788261708681831956450731070, 4.25996934684252686182441763976, 4.69880466218716137446323913348, 6.21562287979366289458103843963, 6.88463793489564166246625751736, 7.52447671513653828914458716255, 8.073364372012518504389811101694
