| L(s) = 1 | − 4·7-s + 2·11-s + 13-s + 2·19-s − 2·23-s − 10·29-s + 4·31-s − 6·37-s + 6·41-s − 8·43-s + 12·47-s + 9·49-s + 14·53-s − 6·59-s + 2·61-s + 4·67-s + 14·73-s − 8·77-s − 12·83-s − 6·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s + 0.277·13-s + 0.458·19-s − 0.417·23-s − 1.85·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s + 1.92·53-s − 0.781·59-s + 0.256·61-s + 0.488·67-s + 1.63·73-s − 0.911·77-s − 1.31·83-s − 0.635·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436009895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436009895\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.74835465323315, −13.39420018050105, −12.80286269820989, −12.41805149948758, −11.92007343562815, −11.43758867787972, −10.82566415692140, −10.28635504309267, −9.792263051786348, −9.406194576647387, −8.926899945492695, −8.450561293406494, −7.665649627285254, −7.159897594432936, −6.741472161538599, −6.176966781921879, −5.681921566705014, −5.239690981192201, −4.248886115512580, −3.796191983084492, −3.417163604442531, −2.683703263417662, −2.060604350634859, −1.182221017056570, −0.4060988425361190,
0.4060988425361190, 1.182221017056570, 2.060604350634859, 2.683703263417662, 3.417163604442531, 3.796191983084492, 4.248886115512580, 5.239690981192201, 5.681921566705014, 6.176966781921879, 6.741472161538599, 7.159897594432936, 7.665649627285254, 8.450561293406494, 8.926899945492695, 9.406194576647387, 9.792263051786348, 10.28635504309267, 10.82566415692140, 11.43758867787972, 11.92007343562815, 12.41805149948758, 12.80286269820989, 13.39420018050105, 13.74835465323315
