| L(s) = 1 | − 2-s − 2.78·3-s + 4-s − 4.36·5-s + 2.78·6-s − 3.26·7-s − 8-s + 4.74·9-s + 4.36·10-s − 3.11·11-s − 2.78·12-s + 2.74·13-s + 3.26·14-s + 12.1·15-s + 16-s − 0.733·17-s − 4.74·18-s + 3.29·19-s − 4.36·20-s + 9.08·21-s + 3.11·22-s − 3.92·23-s + 2.78·24-s + 14.0·25-s − 2.74·26-s − 4.85·27-s − 3.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s − 1.95·5-s + 1.13·6-s − 1.23·7-s − 0.353·8-s + 1.58·9-s + 1.38·10-s − 0.939·11-s − 0.803·12-s + 0.761·13-s + 0.872·14-s + 3.13·15-s + 0.250·16-s − 0.177·17-s − 1.11·18-s + 0.756·19-s − 0.976·20-s + 1.98·21-s + 0.664·22-s − 0.819·23-s + 0.568·24-s + 2.81·25-s − 0.538·26-s − 0.933·27-s − 0.617·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08739315009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08739315009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 4.36T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 0.733T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 0.134T + 29T^{2} \) |
| 31 | \( 1 + 0.0430T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 + 9.77T + 53T^{2} \) |
| 59 | \( 1 + 0.605T + 59T^{2} \) |
| 61 | \( 1 + 9.60T + 61T^{2} \) |
| 67 | \( 1 - 2.80T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 4.01T + 79T^{2} \) |
| 83 | \( 1 - 3.72T + 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
| 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
−7.78783540333123452868091140341, −7.12635021628925819665632581036, −6.54017829751689290272932624067, −5.93683650999732313106554131853, −5.11864318211177607944738510713, −4.28982684768275160933915682464, −3.56333130009665154814897612064, −2.82397117364083950199660331404, −1.10813460644385581594682301341, −0.21348940749198932546394205336,
0.21348940749198932546394205336, 1.10813460644385581594682301341, 2.82397117364083950199660331404, 3.56333130009665154814897612064, 4.28982684768275160933915682464, 5.11864318211177607944738510713, 5.93683650999732313106554131853, 6.54017829751689290272932624067, 7.12635021628925819665632581036, 7.78783540333123452868091140341
