L(s) = 1 | − 2-s + 2.67·3-s + 4-s − 5-s − 2.67·6-s − 2.17·7-s − 8-s + 4.16·9-s + 10-s + 1.23·11-s + 2.67·12-s − 3.34·13-s + 2.17·14-s − 2.67·15-s + 16-s − 2.59·17-s − 4.16·18-s + 0.00342·19-s − 20-s − 5.82·21-s − 1.23·22-s + 6.66·23-s − 2.67·24-s + 25-s + 3.34·26-s + 3.12·27-s − 2.17·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.54·3-s + 0.5·4-s − 0.447·5-s − 1.09·6-s − 0.822·7-s − 0.353·8-s + 1.38·9-s + 0.316·10-s + 0.373·11-s + 0.772·12-s − 0.926·13-s + 0.581·14-s − 0.691·15-s + 0.250·16-s − 0.629·17-s − 0.982·18-s + 0.000785·19-s − 0.223·20-s − 1.27·21-s − 0.263·22-s + 1.38·23-s − 0.546·24-s + 0.200·25-s + 0.655·26-s + 0.601·27-s − 0.411·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 0.00342T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 0.482T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 7.98T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 - 7.10T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 0.836T + 67T^{2} \) |
| 71 | \( 1 + 7.08T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 0.817T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 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
−7.66832938368627539075863500043, −7.34912451758697894019143983634, −6.71011602471197563104521560542, −5.73584094868619274384891385735, −4.53554585668989767366903352884, −3.80628501986763927922004435616, −2.95954238544527135806515982069, −2.52689990975562803242709353791, −1.43678832246623296169117998096, 0,
1.43678832246623296169117998096, 2.52689990975562803242709353791, 2.95954238544527135806515982069, 3.80628501986763927922004435616, 4.53554585668989767366903352884, 5.73584094868619274384891385735, 6.71011602471197563104521560542, 7.34912451758697894019143983634, 7.66832938368627539075863500043