| L(s) = 1 | − 7-s + 2·11-s − 4·13-s + 2·17-s + 2·19-s − 4·23-s + 2·29-s + 6·31-s + 6·37-s − 6·41-s − 4·43-s + 49-s + 8·53-s − 10·61-s − 12·67-s − 14·71-s − 4·73-s − 2·77-s + 8·79-s − 12·83-s + 14·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 0.371·29-s + 1.07·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 1.09·53-s − 1.28·61-s − 1.46·67-s − 1.66·71-s − 0.468·73-s − 0.227·77-s + 0.900·79-s − 1.31·83-s + 1.48·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.51441193847317, −15.10768768795183, −14.55553139571265, −14.03784695695784, −13.52462989115435, −12.98172780397825, −12.22085983961055, −11.85464553640614, −11.61383322287877, −10.50343765091557, −10.17536967021672, −9.677026192541882, −9.106564992962000, −8.473360366901686, −7.763241829890090, −7.334008173665977, −6.620323462954724, −6.087561465276358, −5.458032730327447, −4.654510092649834, −4.220898503942719, −3.283468840661322, −2.798637233771874, −1.913298887304845, −1.038958915689177, 0,
1.038958915689177, 1.913298887304845, 2.798637233771874, 3.283468840661322, 4.220898503942719, 4.654510092649834, 5.458032730327447, 6.087561465276358, 6.620323462954724, 7.334008173665977, 7.763241829890090, 8.473360366901686, 9.106564992962000, 9.677026192541882, 10.17536967021672, 10.50343765091557, 11.61383322287877, 11.85464553640614, 12.22085983961055, 12.98172780397825, 13.52462989115435, 14.03784695695784, 14.55553139571265, 15.10768768795183, 15.51441193847317
