| L(s) = 1 | + 3-s + 3.29·5-s + 2.97·7-s + 9-s − 4.64·11-s + 13-s + 3.29·15-s + 2.52·17-s − 2.18·19-s + 2.97·21-s + 7.18·23-s + 5.85·25-s + 27-s + 8.39·29-s + 5.60·31-s − 4.64·33-s + 9.81·35-s − 5.55·37-s + 39-s + 7.09·41-s − 6.08·43-s + 3.29·45-s + 4.27·47-s + 1.86·49-s + 2.52·51-s − 2.30·53-s − 15.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.47·5-s + 1.12·7-s + 0.333·9-s − 1.39·11-s + 0.277·13-s + 0.850·15-s + 0.612·17-s − 0.501·19-s + 0.649·21-s + 1.49·23-s + 1.17·25-s + 0.192·27-s + 1.55·29-s + 1.00·31-s − 0.808·33-s + 1.65·35-s − 0.913·37-s + 0.160·39-s + 1.10·41-s − 0.927·43-s + 0.491·45-s + 0.623·47-s + 0.266·49-s + 0.353·51-s − 0.316·53-s − 2.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.408493758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408493758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−7.80359734752269335532063286345, −6.99255954838807803595935798749, −6.28064447916536121935468353747, −5.44948168461746890821917104696, −5.02789317303616230236565702805, −4.38643547281137677303637297111, −3.03207844459610508290275392287, −2.62128288195557941032907584465, −1.77546907312426880405839137051, −1.03914288624208137908835713955,
1.03914288624208137908835713955, 1.77546907312426880405839137051, 2.62128288195557941032907584465, 3.03207844459610508290275392287, 4.38643547281137677303637297111, 5.02789317303616230236565702805, 5.44948168461746890821917104696, 6.28064447916536121935468353747, 6.99255954838807803595935798749, 7.80359734752269335532063286345
