| L(s) = 1 | − 2-s + 4-s + 4.06·5-s + 1.57·7-s − 8-s − 4.06·10-s + 5.58·11-s − 4.93·13-s − 1.57·14-s + 16-s − 7.87·17-s + 7.88·19-s + 4.06·20-s − 5.58·22-s + 0.940·23-s + 11.5·25-s + 4.93·26-s + 1.57·28-s + 0.204·29-s − 0.902·31-s − 32-s + 7.87·34-s + 6.39·35-s + 0.171·37-s − 7.88·38-s − 4.06·40-s + 4.80·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.81·5-s + 0.594·7-s − 0.353·8-s − 1.28·10-s + 1.68·11-s − 1.36·13-s − 0.420·14-s + 0.250·16-s − 1.90·17-s + 1.80·19-s + 0.909·20-s − 1.19·22-s + 0.196·23-s + 2.30·25-s + 0.968·26-s + 0.297·28-s + 0.0380·29-s − 0.162·31-s − 0.176·32-s + 1.35·34-s + 1.08·35-s + 0.0281·37-s − 1.27·38-s − 0.643·40-s + 0.750·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.583557344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583557344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 + 7.87T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 - 0.940T + 23T^{2} \) |
| 29 | \( 1 - 0.204T + 29T^{2} \) |
| 31 | \( 1 + 0.902T + 31T^{2} \) |
| 37 | \( 1 - 0.171T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 5.30T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 0.418T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 0.739T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 0.822T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75922907664227746686867682396, −7.09912514376653745947102249789, −6.47483801366608224728503816684, −5.98198671466903373487192541655, −5.03086476507244210685497877306, −4.56252673629732896207623451157, −3.22592484069271954338297304420, −2.28012869115414563546691402856, −1.78857982175113800888951905912, −0.930622657013872527449955177953,
0.930622657013872527449955177953, 1.78857982175113800888951905912, 2.28012869115414563546691402856, 3.22592484069271954338297304420, 4.56252673629732896207623451157, 5.03086476507244210685497877306, 5.98198671466903373487192541655, 6.47483801366608224728503816684, 7.09912514376653745947102249789, 7.75922907664227746686867682396
