| L(s) = 1 | − 2-s − 1.61·3-s + 4-s + 0.900·5-s + 1.61·6-s − 2.89·7-s − 8-s − 0.397·9-s − 0.900·10-s − 11-s − 1.61·12-s − 1.81·13-s + 2.89·14-s − 1.45·15-s + 16-s − 5.57·17-s + 0.397·18-s + 0.900·20-s + 4.66·21-s + 22-s + 5.80·23-s + 1.61·24-s − 4.18·25-s + 1.81·26-s + 5.48·27-s − 2.89·28-s + 5.78·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.931·3-s + 0.5·4-s + 0.402·5-s + 0.658·6-s − 1.09·7-s − 0.353·8-s − 0.132·9-s − 0.284·10-s − 0.301·11-s − 0.465·12-s − 0.502·13-s + 0.773·14-s − 0.375·15-s + 0.250·16-s − 1.35·17-s + 0.0935·18-s + 0.201·20-s + 1.01·21-s + 0.213·22-s + 1.20·23-s + 0.329·24-s − 0.837·25-s + 0.355·26-s + 1.05·27-s − 0.546·28-s + 1.07·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - 0.900T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 5.57T + 17T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 - 5.78T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 + 5.68T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.35740223845143285909297316563, −6.65364020689953638121569713444, −6.31738246039844915115770474159, −5.55281485647435082885843456862, −4.91246914272481965017310889184, −3.89694127385843848501740781997, −2.80044726106292306184389583334, −2.28201626860287432098423341959, −0.863700123358536946702857790913, 0,
0.863700123358536946702857790913, 2.28201626860287432098423341959, 2.80044726106292306184389583334, 3.89694127385843848501740781997, 4.91246914272481965017310889184, 5.55281485647435082885843456862, 6.31738246039844915115770474159, 6.65364020689953638121569713444, 7.35740223845143285909297316563
