| L(s) = 1 | − 3.46i·5-s − 4.89·7-s + 5.65i·11-s − 6.99·25-s + 10.3i·29-s + 4.89·31-s + 16.9i·35-s + 16.9·49-s + 3.46i·53-s + 19.5·55-s + 11.3i·59-s − 14·73-s − 27.7i·77-s − 14.6·79-s + 5.65i·83-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s − 1.85·7-s + 1.70i·11-s − 1.39·25-s + 1.92i·29-s + 0.879·31-s + 2.86i·35-s + 2.42·49-s + 0.475i·53-s + 2.64·55-s + 1.47i·59-s − 1.63·73-s − 3.15i·77-s − 1.65·79-s + 0.620i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6113337949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6113337949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.834826114698587534543119454293, −9.222895353953833908400564058776, −8.647542678586343367902078595366, −7.40667472428260778784304034226, −6.74448827085627929599491985982, −5.71420689860067088581135212892, −4.79796716389535512448469763249, −4.01531609561052349397403917776, −2.77832886122947990504311146832, −1.32259295959350422694783040498,
0.27470739761786917092674999876, 2.66065818697128942592541184597, 3.16332462795423637705025466474, 3.95246053036754736875594414095, 5.82416912659842384510228385358, 6.25925586591240642926255581502, 6.85948796375046758706540448604, 7.86012570261058891342429343764, 8.852757897691642382572335109849, 9.874731862236843750984012965953
