L(s) = 1 | − 2-s − 1.68·3-s + 4-s + 5-s + 1.68·6-s + 7-s − 8-s − 0.154·9-s − 10-s − 1.68·12-s + 0.0425·13-s − 14-s − 1.68·15-s + 16-s − 4.72·17-s + 0.154·18-s − 1.71·19-s + 20-s − 1.68·21-s − 6.95·23-s + 1.68·24-s + 25-s − 0.0425·26-s + 5.32·27-s + 28-s − 1.34·29-s + 1.68·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.973·3-s + 0.5·4-s + 0.447·5-s + 0.688·6-s + 0.377·7-s − 0.353·8-s − 0.0514·9-s − 0.316·10-s − 0.486·12-s + 0.0118·13-s − 0.267·14-s − 0.435·15-s + 0.250·16-s − 1.14·17-s + 0.0363·18-s − 0.393·19-s + 0.223·20-s − 0.368·21-s − 1.45·23-s + 0.344·24-s + 0.200·25-s − 0.00834·26-s + 1.02·27-s + 0.188·28-s − 0.249·29-s + 0.307·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 13 | \( 1 - 0.0425T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 - 6.42T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.98T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.48742823863045511421551248761, −6.55268186735228865560216335618, −6.18302464971627447455047529576, −5.61001726170900092686766056149, −4.69624973263199464701451589625, −4.08962249234853594357072048580, −2.72941418065034483327029291135, −2.10491550086078595014255429194, −1.02419502154724618324195233033, 0,
1.02419502154724618324195233033, 2.10491550086078595014255429194, 2.72941418065034483327029291135, 4.08962249234853594357072048580, 4.69624973263199464701451589625, 5.61001726170900092686766056149, 6.18302464971627447455047529576, 6.55268186735228865560216335618, 7.48742823863045511421551248761