L(s) = 1 | − 2-s + 4-s − 5-s − 4.37·7-s − 8-s + 10-s − 2.37·11-s + 6.74·13-s + 4.37·14-s + 16-s − 0.372·17-s − 2·19-s − 20-s + 2.37·22-s + 4.74·23-s + 25-s − 6.74·26-s − 4.37·28-s + 9.11·29-s − 8.37·31-s − 32-s + 0.372·34-s + 4.37·35-s + 37-s + 2·38-s + 40-s + 0.372·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.65·7-s − 0.353·8-s + 0.316·10-s − 0.715·11-s + 1.87·13-s + 1.16·14-s + 0.250·16-s − 0.0902·17-s − 0.458·19-s − 0.223·20-s + 0.505·22-s + 0.989·23-s + 0.200·25-s − 1.32·26-s − 0.826·28-s + 1.69·29-s − 1.50·31-s − 0.176·32-s + 0.0638·34-s + 0.739·35-s + 0.164·37-s + 0.324·38-s + 0.158·40-s + 0.0581·41-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 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 41 | \( 1 - 0.372T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.469771364309071874645677588900, −7.54178772478744183070947367161, −6.72946777531976917641707384425, −6.27244862401280591480099986100, −5.44074539024667872047798971939, −4.12293683094400550069460105472, −3.33173661703590294969949076592, −2.68561958565320940164590418829, −1.17192536847463201195269271009, 0,
1.17192536847463201195269271009, 2.68561958565320940164590418829, 3.33173661703590294969949076592, 4.12293683094400550069460105472, 5.44074539024667872047798971939, 6.27244862401280591480099986100, 6.72946777531976917641707384425, 7.54178772478744183070947367161, 8.469771364309071874645677588900