| L(s) = 1 | − 2-s − 2.45·3-s + 4-s − 4.36·5-s + 2.45·6-s + 2.66·7-s − 8-s + 3.03·9-s + 4.36·10-s + 11-s − 2.45·12-s + 3.00·13-s − 2.66·14-s + 10.7·15-s + 16-s − 1.13·17-s − 3.03·18-s + 4.93·19-s − 4.36·20-s − 6.53·21-s − 22-s + 2.45·23-s + 2.45·24-s + 14.0·25-s − 3.00·26-s − 0.0953·27-s + 2.66·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.41·3-s + 0.5·4-s − 1.95·5-s + 1.00·6-s + 1.00·7-s − 0.353·8-s + 1.01·9-s + 1.37·10-s + 0.301·11-s − 0.709·12-s + 0.833·13-s − 0.711·14-s + 2.76·15-s + 0.250·16-s − 0.274·17-s − 0.716·18-s + 1.13·19-s − 0.975·20-s − 1.42·21-s − 0.213·22-s + 0.512·23-s + 0.501·24-s + 2.80·25-s − 0.589·26-s − 0.0183·27-s + 0.502·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6447288497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6447288497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 4.36T + 5T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 - 6.28T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 - 9.98T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 0.0329T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 6.65T + 89T^{2} \) |
| 97 | \( 1 - 3.27T + 97T^{2} \) |
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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
−8.501511042004204533207337835342, −7.66281449231449947914967167648, −6.93154410576883057175982456667, −6.47579985760075304339645727279, −5.26244554040545665770230897695, −4.78313170837950262657482517399, −3.93986068412793044943107178156, −3.02533611964206600480457890496, −1.26684324651358877039410410877, −0.64339941787331981984250051124,
0.64339941787331981984250051124, 1.26684324651358877039410410877, 3.02533611964206600480457890496, 3.93986068412793044943107178156, 4.78313170837950262657482517399, 5.26244554040545665770230897695, 6.47579985760075304339645727279, 6.93154410576883057175982456667, 7.66281449231449947914967167648, 8.501511042004204533207337835342
