| L(s) = 1 | − 2-s + 1.30·3-s + 4-s − 5-s − 1.30·6-s + 3.65·7-s − 8-s − 1.30·9-s + 10-s + 1.60·11-s + 1.30·12-s + 1.38·13-s − 3.65·14-s − 1.30·15-s + 16-s − 3.79·17-s + 1.30·18-s − 7.12·19-s − 20-s + 4.75·21-s − 1.60·22-s + 8.11·23-s − 1.30·24-s + 25-s − 1.38·26-s − 5.60·27-s + 3.65·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.751·3-s + 0.5·4-s − 0.447·5-s − 0.531·6-s + 1.38·7-s − 0.353·8-s − 0.435·9-s + 0.316·10-s + 0.483·11-s + 0.375·12-s + 0.384·13-s − 0.976·14-s − 0.336·15-s + 0.250·16-s − 0.921·17-s + 0.307·18-s − 1.63·19-s − 0.223·20-s + 1.03·21-s − 0.341·22-s + 1.69·23-s − 0.265·24-s + 0.200·25-s − 0.271·26-s − 1.07·27-s + 0.690·28-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 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 + 9.33T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 + 9.72T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 + 8.78T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.476T + 73T^{2} \) |
| 79 | \( 1 + 2.28T + 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−7.972358274882574481397964725650, −7.24368892136373964487450131377, −6.56045931144825517346156995839, −5.61474187457649525330746422543, −4.71612976900040857706755299371, −3.97908419194605925630384879469, −3.09814189822254701737356449279, −2.10999291997850504974609202395, −1.50370217699232515386872655679, 0,
1.50370217699232515386872655679, 2.10999291997850504974609202395, 3.09814189822254701737356449279, 3.97908419194605925630384879469, 4.71612976900040857706755299371, 5.61474187457649525330746422543, 6.56045931144825517346156995839, 7.24368892136373964487450131377, 7.972358274882574481397964725650
