L(s) = 1 | − 2-s − 0.206·3-s + 4-s + 5-s + 0.206·6-s − 1.71·7-s − 8-s − 2.95·9-s − 10-s − 11-s − 0.206·12-s − 7.14·13-s + 1.71·14-s − 0.206·15-s + 16-s + 6.61·17-s + 2.95·18-s + 3.44·19-s + 20-s + 0.353·21-s + 22-s + 0.503·23-s + 0.206·24-s + 25-s + 7.14·26-s + 1.23·27-s − 1.71·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.119·3-s + 0.5·4-s + 0.447·5-s + 0.0844·6-s − 0.646·7-s − 0.353·8-s − 0.985·9-s − 0.316·10-s − 0.301·11-s − 0.0596·12-s − 1.98·13-s + 0.457·14-s − 0.0533·15-s + 0.250·16-s + 1.60·17-s + 0.697·18-s + 0.791·19-s + 0.223·20-s + 0.0772·21-s + 0.213·22-s + 0.104·23-s + 0.0422·24-s + 0.200·25-s + 1.40·26-s + 0.237·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7672057989\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7672057989\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.206T + 3T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 13 | \( 1 + 7.14T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 - 0.503T + 23T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 7.14T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−8.252540045100397320385029092954, −7.46782691022035133345330471264, −7.18162193008178654287740107103, −5.98194168269162490091931946739, −5.57922038678111889822054378305, −4.83804103993711436043396407670, −3.35141372941453960416010413649, −2.85157808800020088141213145531, −1.91227089197938808324933482417, −0.51596464537423928475349764277,
0.51596464537423928475349764277, 1.91227089197938808324933482417, 2.85157808800020088141213145531, 3.35141372941453960416010413649, 4.83804103993711436043396407670, 5.57922038678111889822054378305, 5.98194168269162490091931946739, 7.18162193008178654287740107103, 7.46782691022035133345330471264, 8.252540045100397320385029092954