L(s) = 1 | − 2-s − 1.35·3-s + 4-s − 2.02·5-s + 1.35·6-s + 1.66·7-s − 8-s − 1.17·9-s + 2.02·10-s + 1.39·11-s − 1.35·12-s − 1.66·14-s + 2.73·15-s + 16-s − 2.66·17-s + 1.17·18-s + 19-s − 2.02·20-s − 2.25·21-s − 1.39·22-s + 0.569·23-s + 1.35·24-s − 0.894·25-s + 5.64·27-s + 1.66·28-s + 4.44·29-s − 2.73·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.779·3-s + 0.5·4-s − 0.906·5-s + 0.551·6-s + 0.631·7-s − 0.353·8-s − 0.391·9-s + 0.640·10-s + 0.419·11-s − 0.389·12-s − 0.446·14-s + 0.706·15-s + 0.250·16-s − 0.645·17-s + 0.277·18-s + 0.229·19-s − 0.453·20-s − 0.492·21-s − 0.296·22-s + 0.118·23-s + 0.275·24-s − 0.178·25-s + 1.08·27-s + 0.315·28-s + 0.825·29-s − 0.499·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 23 | \( 1 - 0.569T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 0.328T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 - 0.970T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 + 2.33T + 89T^{2} \) |
| 97 | \( 1 - 0.797T + 97T^{2} \) |
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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
−7.62170514798876730613769637304, −7.13873934827900498796350194179, −6.31463799314468953211660621836, −5.65001606778937968038456393647, −4.82357460622698352983889089540, −4.09192454701839846931002383295, −3.16322070827788570521858752449, −2.10664587686515664380391057911, −0.985715178484143966737154026249, 0,
0.985715178484143966737154026249, 2.10664587686515664380391057911, 3.16322070827788570521858752449, 4.09192454701839846931002383295, 4.82357460622698352983889089540, 5.65001606778937968038456393647, 6.31463799314468953211660621836, 7.13873934827900498796350194179, 7.62170514798876730613769637304