| L(s) = 1 | − 2-s − 2.35·3-s + 4-s − 1.55·5-s + 2.35·6-s − 2.49·7-s − 8-s + 2.55·9-s + 1.55·10-s − 2.29·11-s − 2.35·12-s + 2.49·14-s + 3.66·15-s + 16-s − 7.74·17-s − 2.55·18-s + 19-s − 1.55·20-s + 5.87·21-s + 2.29·22-s + 0.0217·23-s + 2.35·24-s − 2.58·25-s + 1.04·27-s − 2.49·28-s + 2.29·29-s − 3.66·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s − 0.695·5-s + 0.962·6-s − 0.942·7-s − 0.353·8-s + 0.851·9-s + 0.491·10-s − 0.692·11-s − 0.680·12-s + 0.666·14-s + 0.946·15-s + 0.250·16-s − 1.87·17-s − 0.602·18-s + 0.229·19-s − 0.347·20-s + 1.28·21-s + 0.489·22-s + 0.00453·23-s + 0.481·24-s − 0.516·25-s + 0.201·27-s − 0.471·28-s + 0.426·29-s − 0.669·30-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 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 23 | \( 1 - 0.0217T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 - 0.939T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.814T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.65432784550620120817712111415, −6.80349562710393178648663268651, −6.40454141684016554515765399059, −5.76606015708736389040020714085, −4.83319031061953767661252582656, −4.17531272817241524506847927497, −3.09500185747662114203027021268, −2.20294266736568069362544404883, −0.71413681898900800020822054658, 0,
0.71413681898900800020822054658, 2.20294266736568069362544404883, 3.09500185747662114203027021268, 4.17531272817241524506847927497, 4.83319031061953767661252582656, 5.76606015708736389040020714085, 6.40454141684016554515765399059, 6.80349562710393178648663268651, 7.65432784550620120817712111415
