| L(s) = 1 | − 3.31·3-s − 2.56·5-s + 1.56·7-s + 7.95·9-s + 2.12·11-s − 6.14·13-s + 8.48·15-s − 7.33·17-s − 3.28·19-s − 5.16·21-s − 2.25·23-s + 1.56·25-s − 16.4·27-s + 3.64·29-s + 3.04·31-s − 7.02·33-s − 4.00·35-s − 7.09·37-s + 20.3·39-s + 8.18·41-s + 0.642·43-s − 20.3·45-s + 11.1·47-s − 4.56·49-s + 24.2·51-s − 6.50·53-s − 5.43·55-s + ⋯ |
| L(s) = 1 | − 1.91·3-s − 1.14·5-s + 0.590·7-s + 2.65·9-s + 0.639·11-s − 1.70·13-s + 2.18·15-s − 1.77·17-s − 0.753·19-s − 1.12·21-s − 0.469·23-s + 0.312·25-s − 3.15·27-s + 0.676·29-s + 0.546·31-s − 1.22·33-s − 0.676·35-s − 1.16·37-s + 3.25·39-s + 1.27·41-s + 0.0979·43-s − 3.03·45-s + 1.62·47-s − 0.651·49-s + 3.39·51-s − 0.893·53-s − 0.732·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4006612118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4006612118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 + 3.28T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 7.09T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 0.642T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 6.50T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 7.48T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 + 6.79T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−9.917000101183522942259720837626, −8.798809264318329186274066403498, −7.71789568478038288970367383912, −6.99591260370803906735596947073, −6.43490050763877004686676896096, −5.31211933737189443654712177670, −4.44850742426320988914006606571, −4.20268906604914049557081903125, −2.11262817308438600543630058062, −0.49181182522703575664438345486,
0.49181182522703575664438345486, 2.11262817308438600543630058062, 4.20268906604914049557081903125, 4.44850742426320988914006606571, 5.31211933737189443654712177670, 6.43490050763877004686676896096, 6.99591260370803906735596947073, 7.71789568478038288970367383912, 8.798809264318329186274066403498, 9.917000101183522942259720837626
