| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 4·11-s − 13-s + 16-s + 4·17-s − 20-s + 4·22-s − 2·23-s + 25-s − 26-s + 29-s + 4·31-s + 32-s + 4·34-s + 2·37-s − 40-s + 4·43-s + 4·44-s − 2·46-s − 7·49-s + 50-s − 52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s − 0.196·26-s + 0.185·29-s + 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.328·37-s − 0.158·40-s + 0.609·43-s + 0.603·44-s − 0.294·46-s − 49-s + 0.141·50-s − 0.138·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.149503918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.149503918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.89728329276228, −14.44093148687371, −14.01293190090984, −13.54071993306900, −12.72522343864599, −12.35064259113274, −11.93847078185560, −11.37520632552948, −10.99728973791424, −10.10977617374216, −9.767732931034814, −9.120617613822378, −8.331783600626399, −7.939259605938023, −7.243473615177966, −6.664852814939581, −6.219505767238465, −5.474845153354829, −4.930570683901356, −4.137329687850969, −3.811253930414095, −3.090069451997605, −2.369618597883099, −1.460665647656543, −0.7191884788527298,
0.7191884788527298, 1.460665647656543, 2.369618597883099, 3.090069451997605, 3.811253930414095, 4.137329687850969, 4.930570683901356, 5.474845153354829, 6.219505767238465, 6.664852814939581, 7.243473615177966, 7.939259605938023, 8.331783600626399, 9.120617613822378, 9.767732931034814, 10.10977617374216, 10.99728973791424, 11.37520632552948, 11.93847078185560, 12.35064259113274, 12.72522343864599, 13.54071993306900, 14.01293190090984, 14.44093148687371, 14.89728329276228
