| L(s) = 1 | + 1.52·2-s − 2.99·3-s + 0.320·4-s + 2.94·5-s − 4.56·6-s − 2.55·8-s + 5.97·9-s + 4.48·10-s − 2.37·11-s − 0.960·12-s − 8.82·15-s − 4.53·16-s − 5.36·17-s + 9.09·18-s + 5.35·19-s + 0.944·20-s − 3.61·22-s + 2.79·23-s + 7.66·24-s + 3.67·25-s − 8.89·27-s + 0.585·29-s − 13.4·30-s − 8.33·31-s − 1.79·32-s + 7.10·33-s − 8.18·34-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 1.72·3-s + 0.160·4-s + 1.31·5-s − 1.86·6-s − 0.904·8-s + 1.99·9-s + 1.41·10-s − 0.714·11-s − 0.277·12-s − 2.27·15-s − 1.13·16-s − 1.30·17-s + 2.14·18-s + 1.22·19-s + 0.211·20-s − 0.770·22-s + 0.581·23-s + 1.56·24-s + 0.735·25-s − 1.71·27-s + 0.108·29-s − 2.45·30-s − 1.49·31-s − 0.317·32-s + 1.23·33-s − 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651037551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651037551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 2.79T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + 8.33T + 31T^{2} \) |
| 37 | \( 1 - 0.675T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.537T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 2.78T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 9.90T + 97T^{2} \) |
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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
−7.26138201202715453414376614667, −6.81373115629717803931881524655, −6.06118291377001326893631882900, −5.56042649868516237757970416148, −5.22698135318196425951299014946, −4.65841434520551046457595031807, −3.78049562858760684073236036432, −2.69712417121931393391479766344, −1.80265606919862304755675059697, −0.57840259240956066913326526817,
0.57840259240956066913326526817, 1.80265606919862304755675059697, 2.69712417121931393391479766344, 3.78049562858760684073236036432, 4.65841434520551046457595031807, 5.22698135318196425951299014946, 5.56042649868516237757970416148, 6.06118291377001326893631882900, 6.81373115629717803931881524655, 7.26138201202715453414376614667
