| L(s) = 1 | + 4·7-s − 2·11-s − 4·13-s + 17-s − 4·19-s − 8·23-s − 8·29-s + 2·37-s + 4·41-s − 6·43-s − 12·47-s + 9·49-s + 14·53-s + 2·61-s − 2·67-s − 14·71-s − 2·73-s − 8·77-s + 4·79-s − 16·83-s + 6·89-s − 16·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s − 1.48·29-s + 0.328·37-s + 0.624·41-s − 0.914·43-s − 1.75·47-s + 9/7·49-s + 1.92·53-s + 0.256·61-s − 0.244·67-s − 1.66·71-s − 0.234·73-s − 0.911·77-s + 0.450·79-s − 1.75·83-s + 0.635·89-s − 1.67·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.557176151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557176151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.89491960818887, −14.60549221758796, −14.32939949034786, −13.45398372429749, −13.06546372846341, −12.43376864180273, −11.70419551177314, −11.57702641755502, −10.85997415515032, −10.14914028730110, −9.994746833667725, −9.058601640159228, −8.487535257583545, −7.939486081548944, −7.581623087311314, −7.019093973099796, −6.058916022536027, −5.587220751718354, −4.926082551074955, −4.451135268196971, −3.838731695295986, −2.864779675595173, −1.948414513887697, −1.845011319139920, −0.4508583406511639,
0.4508583406511639, 1.845011319139920, 1.948414513887697, 2.864779675595173, 3.838731695295986, 4.451135268196971, 4.926082551074955, 5.587220751718354, 6.058916022536027, 7.019093973099796, 7.581623087311314, 7.939486081548944, 8.487535257583545, 9.058601640159228, 9.994746833667725, 10.14914028730110, 10.85997415515032, 11.57702641755502, 11.70419551177314, 12.43376864180273, 13.06546372846341, 13.45398372429749, 14.32939949034786, 14.60549221758796, 14.89491960818887
