L(s) = 1 | − 2-s + 4-s + 5-s + 2.91·7-s − 8-s − 10-s − 6.51·11-s − 0.813·13-s − 2.91·14-s + 16-s + 2.51·17-s + 0.406·19-s + 20-s + 6.51·22-s − 5.02·23-s + 25-s + 0.813·26-s + 2.91·28-s + 5.32·29-s − 8.75·31-s − 32-s − 2.51·34-s + 2.91·35-s − 37-s − 0.406·38-s − 40-s − 6.34·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.10·7-s − 0.353·8-s − 0.316·10-s − 1.96·11-s − 0.225·13-s − 0.779·14-s + 0.250·16-s + 0.608·17-s + 0.0933·19-s + 0.223·20-s + 1.38·22-s − 1.04·23-s + 0.200·25-s + 0.159·26-s + 0.551·28-s + 0.988·29-s − 1.57·31-s − 0.176·32-s − 0.430·34-s + 0.493·35-s − 0.164·37-s − 0.0659·38-s − 0.158·40-s − 0.990·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 + 6.51T + 11T^{2} \) |
| 13 | \( 1 + 0.813T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 0.406T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 + 7.32T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−8.210328469619806186309293623023, −7.69579993441764150048228477360, −7.06003618243218109488299685193, −5.84452576302961133975493979749, −5.34085452700740260712226011633, −4.58875985804778165614455517511, −3.21853516905563634382777127551, −2.31004511769648227849935915419, −1.53138459019457292030731447648, 0,
1.53138459019457292030731447648, 2.31004511769648227849935915419, 3.21853516905563634382777127551, 4.58875985804778165614455517511, 5.34085452700740260712226011633, 5.84452576302961133975493979749, 7.06003618243218109488299685193, 7.69579993441764150048228477360, 8.210328469619806186309293623023