| L(s) = 1 | − 0.157·3-s − 0.360·5-s − 0.00399·7-s − 2.97·9-s + 4.38·11-s + 3.05·13-s + 0.0568·15-s + 4.57·17-s − 6.57·19-s + 0.000630·21-s + 5.45·23-s − 4.86·25-s + 0.941·27-s + 5.93·29-s − 0.0296·31-s − 0.691·33-s + 0.00144·35-s − 5.18·37-s − 0.482·39-s − 8.74·41-s + 7.53·43-s + 1.07·45-s + 8.87·47-s − 6.99·49-s − 0.720·51-s + 0.242·53-s − 1.58·55-s + ⋯ |
| L(s) = 1 | − 0.0910·3-s − 0.161·5-s − 0.00151·7-s − 0.991·9-s + 1.32·11-s + 0.848·13-s + 0.0146·15-s + 1.10·17-s − 1.50·19-s + 0.000137·21-s + 1.13·23-s − 0.973·25-s + 0.181·27-s + 1.10·29-s − 0.00531·31-s − 0.120·33-s + 0.000243·35-s − 0.852·37-s − 0.0772·39-s − 1.36·41-s + 1.14·43-s + 0.160·45-s + 1.29·47-s − 0.999·49-s − 0.100·51-s + 0.0332·53-s − 0.213·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801312017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801312017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.157T + 3T^{2} \) |
| 5 | \( 1 + 0.360T + 5T^{2} \) |
| 7 | \( 1 + 0.00399T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 0.0296T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 - 8.87T + 47T^{2} \) |
| 53 | \( 1 - 0.242T + 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 - 6.68T + 73T^{2} \) |
| 79 | \( 1 - 5.37T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.601643360749742040179411138471, −7.83217355695360873782691091401, −6.80967372086981597734936687326, −6.26101416597144057857578222836, −5.61106051342245872165501977201, −4.63070004822930152933013677593, −3.75332218667732650660404096742, −3.12572309711237878212989922953, −1.89740743076094930938196183011, −0.790130646697262782584413957583,
0.790130646697262782584413957583, 1.89740743076094930938196183011, 3.12572309711237878212989922953, 3.75332218667732650660404096742, 4.63070004822930152933013677593, 5.61106051342245872165501977201, 6.26101416597144057857578222836, 6.80967372086981597734936687326, 7.83217355695360873782691091401, 8.601643360749742040179411138471
