| L(s) = 1 | + 3.26·3-s + 5-s − 1.62·7-s + 7.64·9-s + 3.45·11-s + 3.89·13-s + 3.26·15-s − 1.19·17-s − 0.150·19-s − 5.30·21-s − 8.16·23-s + 25-s + 15.1·27-s − 5.82·29-s + 8.17·31-s + 11.2·33-s − 1.62·35-s − 37-s + 12.7·39-s + 7.37·41-s + 6.34·43-s + 7.64·45-s − 7.46·47-s − 4.35·49-s − 3.89·51-s + 5.07·53-s + 3.45·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 0.447·5-s − 0.614·7-s + 2.54·9-s + 1.04·11-s + 1.08·13-s + 0.842·15-s − 0.289·17-s − 0.0344·19-s − 1.15·21-s − 1.70·23-s + 0.200·25-s + 2.92·27-s − 1.08·29-s + 1.46·31-s + 1.96·33-s − 0.275·35-s − 0.164·37-s + 2.03·39-s + 1.15·41-s + 0.967·43-s + 1.14·45-s − 1.08·47-s − 0.621·49-s − 0.545·51-s + 0.696·53-s + 0.466·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.289305901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.289305901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 - 3.45T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 + 0.150T + 19T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 8.17T + 31T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 + 0.0485T + 59T^{2} \) |
| 61 | \( 1 - 0.702T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 8.30T + 97T^{2} \) |
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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
−8.820499505306985293703206176220, −8.150385841065432720300868476390, −7.43722883120118090705735173636, −6.49497857285711630366510819064, −5.99832558071090340004237318452, −4.38609977023745849862920038312, −3.85809425617929858538175724885, −3.10855971076699676745827006013, −2.17714402193466588055373754660, −1.33520133118360473840888507626,
1.33520133118360473840888507626, 2.17714402193466588055373754660, 3.10855971076699676745827006013, 3.85809425617929858538175724885, 4.38609977023745849862920038312, 5.99832558071090340004237318452, 6.49497857285711630366510819064, 7.43722883120118090705735173636, 8.150385841065432720300868476390, 8.820499505306985293703206176220
