| L(s) = 1 | + 7-s − 3·11-s + 13-s + 3·17-s − 6·19-s − 23-s − 4·29-s − 8·31-s + 3·37-s + 3·41-s + 6·43-s + 12·47-s − 6·49-s − 53-s + 4·59-s + 5·61-s + 71-s + 14·73-s − 3·77-s − 5·79-s − 3·89-s + 91-s − 15·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.208·23-s − 0.742·29-s − 1.43·31-s + 0.493·37-s + 0.468·41-s + 0.914·43-s + 1.75·47-s − 6/7·49-s − 0.137·53-s + 0.520·59-s + 0.640·61-s + 0.118·71-s + 1.63·73-s − 0.341·77-s − 0.562·79-s − 0.317·89-s + 0.104·91-s − 1.52·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(\approx\) |
\(1.688310223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688310223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−14.00137462077022, −13.18546213217797, −12.77354177139219, −12.57517523855213, −11.87953161696091, −11.14540816474350, −10.94692774626408, −10.49267903066869, −9.861487790307015, −9.362137052298285, −8.783153906718357, −8.286096131270701, −7.774853800305030, −7.377047598338443, −6.768337965125742, −6.014667526843849, −5.634502209633264, −5.159404069186957, −4.404831753115366, −3.930019021140240, −3.352423697772912, −2.447024134274614, −2.140828855569887, −1.279900748749564, −0.4189138649704718,
0.4189138649704718, 1.279900748749564, 2.140828855569887, 2.447024134274614, 3.352423697772912, 3.930019021140240, 4.404831753115366, 5.159404069186957, 5.634502209633264, 6.014667526843849, 6.768337965125742, 7.377047598338443, 7.774853800305030, 8.286096131270701, 8.783153906718357, 9.362137052298285, 9.861487790307015, 10.49267903066869, 10.94692774626408, 11.14540816474350, 11.87953161696091, 12.57517523855213, 12.77354177139219, 13.18546213217797, 14.00137462077022
