L(s) = 1 | − 2.23·2-s + 0.618·3-s + 3.00·4-s − 5-s − 1.38·6-s − 7-s − 2.23·8-s − 2.61·9-s + 2.23·10-s + 1.85·12-s + 4.61·13-s + 2.23·14-s − 0.618·15-s − 0.999·16-s + 5.61·17-s + 5.85·18-s − 3.23·19-s − 3.00·20-s − 0.618·21-s − 4.47·23-s − 1.38·24-s + 25-s − 10.3·26-s − 3.47·27-s − 3.00·28-s + 0.854·29-s + 1.38·30-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.356·3-s + 1.50·4-s − 0.447·5-s − 0.564·6-s − 0.377·7-s − 0.790·8-s − 0.872·9-s + 0.707·10-s + 0.535·12-s + 1.28·13-s + 0.597·14-s − 0.159·15-s − 0.249·16-s + 1.36·17-s + 1.37·18-s − 0.742·19-s − 0.670·20-s − 0.134·21-s − 0.932·23-s − 0.282·24-s + 0.200·25-s − 2.02·26-s − 0.668·27-s − 0.566·28-s + 0.158·29-s + 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 + 4.76T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.108849274358393176865383768812, −7.83381181161921086560740097402, −6.68154807611888045784429879910, −6.21384485473736703666626956045, −5.21606975307320167704675252413, −3.88712960122209134110969594404, −3.23856495992822729425301149416, −2.19808259691164166231137285095, −1.13806523522405957672232335676, 0,
1.13806523522405957672232335676, 2.19808259691164166231137285095, 3.23856495992822729425301149416, 3.88712960122209134110969594404, 5.21606975307320167704675252413, 6.21384485473736703666626956045, 6.68154807611888045784429879910, 7.83381181161921086560740097402, 8.108849274358393176865383768812