L(s) = 1 | − 2-s − 1.53·3-s + 4-s − 3.27·5-s + 1.53·6-s + 1.87·7-s − 8-s − 0.641·9-s + 3.27·10-s + 3.02·11-s − 1.53·12-s − 6.24·13-s − 1.87·14-s + 5.02·15-s + 16-s + 3.72·17-s + 0.641·18-s + 7.36·19-s − 3.27·20-s − 2.88·21-s − 3.02·22-s − 23-s + 1.53·24-s + 5.70·25-s + 6.24·26-s + 5.59·27-s + 1.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.886·3-s + 0.5·4-s − 1.46·5-s + 0.626·6-s + 0.710·7-s − 0.353·8-s − 0.213·9-s + 1.03·10-s + 0.913·11-s − 0.443·12-s − 1.73·13-s − 0.502·14-s + 1.29·15-s + 0.250·16-s + 0.903·17-s + 0.151·18-s + 1.68·19-s − 0.731·20-s − 0.629·21-s − 0.645·22-s − 0.208·23-s + 0.313·24-s + 1.14·25-s + 1.22·26-s + 1.07·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5371039934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5371039934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 - 5.85T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 0.455T + 79T^{2} \) |
| 83 | \( 1 + 4.22T + 83T^{2} \) |
| 89 | \( 1 + 7.83T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.88540314255506304188419604035, −7.50477979531046979337626670135, −6.95638235305591189163878076715, −5.95748202212822701251074852019, −5.16500482971761149880253939210, −4.61651057037909583012696556250, −3.60566997722623633350013896104, −2.81687781370666216614178899866, −1.43599582256989650561281208025, −0.47897713540611883080725704616,
0.47897713540611883080725704616, 1.43599582256989650561281208025, 2.81687781370666216614178899866, 3.60566997722623633350013896104, 4.61651057037909583012696556250, 5.16500482971761149880253939210, 5.95748202212822701251074852019, 6.95638235305591189163878076715, 7.50477979531046979337626670135, 7.88540314255506304188419604035