| L(s) = 1 | + 2-s − 2.39·3-s + 4-s − 5-s − 2.39·6-s − 0.390·7-s + 8-s + 2.73·9-s − 10-s − 11-s − 2.39·12-s + 2.13·13-s − 0.390·14-s + 2.39·15-s + 16-s − 6.00·17-s + 2.73·18-s + 4.60·19-s − 20-s + 0.934·21-s − 22-s − 3.04·23-s − 2.39·24-s + 25-s + 2.13·26-s + 0.643·27-s − 0.390·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.38·3-s + 0.5·4-s − 0.447·5-s − 0.977·6-s − 0.147·7-s + 0.353·8-s + 0.910·9-s − 0.316·10-s − 0.301·11-s − 0.691·12-s + 0.593·13-s − 0.104·14-s + 0.618·15-s + 0.250·16-s − 1.45·17-s + 0.643·18-s + 1.05·19-s − 0.223·20-s + 0.203·21-s − 0.213·22-s − 0.634·23-s − 0.488·24-s + 0.200·25-s + 0.419·26-s + 0.123·27-s − 0.0737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 7 | \( 1 + 0.390T + 7T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 + 9.73T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 59 | \( 1 - 4.39T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 6.37T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.20167223724462421880988913682, −6.55210039964446203127127197790, −6.10633550141401590980159668470, −5.33191056740625375617476506069, −4.80208310076614052438715781285, −4.10098779553320634789996376527, −3.30425424297383357034478881454, −2.29386383420148747220771295270, −1.10187937751390798114345708941, 0,
1.10187937751390798114345708941, 2.29386383420148747220771295270, 3.30425424297383357034478881454, 4.10098779553320634789996376527, 4.80208310076614052438715781285, 5.33191056740625375617476506069, 6.10633550141401590980159668470, 6.55210039964446203127127197790, 7.20167223724462421880988913682
