| L(s) = 1 | − 2-s − 3.12·3-s + 4-s + 4.00·5-s + 3.12·6-s − 0.0131·7-s − 8-s + 6.76·9-s − 4.00·10-s + 11-s − 3.12·12-s − 2.70·13-s + 0.0131·14-s − 12.5·15-s + 16-s − 5.79·17-s − 6.76·18-s + 0.854·19-s + 4.00·20-s + 0.0412·21-s − 22-s − 5.14·23-s + 3.12·24-s + 11.0·25-s + 2.70·26-s − 11.7·27-s − 0.0131·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.80·3-s + 0.5·4-s + 1.78·5-s + 1.27·6-s − 0.00498·7-s − 0.353·8-s + 2.25·9-s − 1.26·10-s + 0.301·11-s − 0.902·12-s − 0.751·13-s + 0.00352·14-s − 3.22·15-s + 0.250·16-s − 1.40·17-s − 1.59·18-s + 0.195·19-s + 0.894·20-s + 0.00899·21-s − 0.213·22-s − 1.07·23-s + 0.638·24-s + 2.20·25-s + 0.531·26-s − 2.26·27-s − 0.00249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8375523055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8375523055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 7 | \( 1 + 0.0131T + 7T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 + 2.67T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 2.64T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 4.69T + 97T^{2} \) |
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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
−8.519538880682327591058106155787, −7.34452044919769471930194685573, −6.64077920389180982804735650471, −6.30021228548314099836023472772, −5.52424071583843784793250307869, −5.07973451058105038603534110545, −4.05960820250414339986234014131, −2.32923079245672032710539064246, −1.78381926258728836426098155402, −0.62751641799534355550778049459,
0.62751641799534355550778049459, 1.78381926258728836426098155402, 2.32923079245672032710539064246, 4.05960820250414339986234014131, 5.07973451058105038603534110545, 5.52424071583843784793250307869, 6.30021228548314099836023472772, 6.64077920389180982804735650471, 7.34452044919769471930194685573, 8.519538880682327591058106155787
