L(s) = 1 | + 8.52i·3-s + 18.2i·5-s + 13.8i·7-s − 45.6·9-s − 60.8i·11-s + 61.8·13-s − 155.·15-s + (69.2 + 10.7i)17-s − 40.0·19-s − 118.·21-s − 4.88i·23-s − 208.·25-s − 158. i·27-s + 113. i·29-s + 95.1i·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + 1.63i·5-s + 0.749i·7-s − 1.69·9-s − 1.66i·11-s + 1.31·13-s − 2.68·15-s + (0.988 + 0.153i)17-s − 0.483·19-s − 1.22·21-s − 0.0443i·23-s − 1.67·25-s − 1.13i·27-s + 0.724i·29-s + 0.551i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.117044 + 1.51860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117044 + 1.51860i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-69.2 - 10.7i)T \) |
good | 3 | \( 1 - 8.52iT - 27T^{2} \) |
| 5 | \( 1 - 18.2iT - 125T^{2} \) |
| 7 | \( 1 - 13.8iT - 343T^{2} \) |
| 11 | \( 1 + 60.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 61.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 40.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.88iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 113. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 95.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 273. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 446. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 27.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 266.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 502. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 724. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 188. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 48.3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 50.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3iT - 9.12e5T^{2} \) |
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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
−13.59425980682325903668440097420, −11.68846355760174908505275090162, −10.83829323919932253192927436805, −10.48696179994705902328067038344, −9.175086423898327236045514886841, −8.252548746551240338377505165413, −6.34142348563780710967914527268, −5.54583112191308759798444194813, −3.64993557346630131539833307498, −3.06683052167864925356585821919,
0.823349142759307950569335209607, 1.77445614502147435550451406196, 4.24030677149390960962996021225, 5.66883136126271393592288773834, 6.99226067915444419689384097529, 7.896968148191090168408693384060, 8.769241921716423703232518565157, 10.10825337644294320039660460715, 11.81968792946952053061492536457, 12.36791717669334295772223052056