| L(s) = 1 | + 4.24·5-s − 6i·13-s − 7.07i·17-s + 12.9·25-s − 4.24·29-s + 12i·37-s − 1.41i·41-s + 7·49-s − 12.7·53-s + 12i·61-s − 25.4i·65-s + 16·73-s − 30i·85-s + 18.3i·89-s − 8·97-s + ⋯ |
| L(s) = 1 | + 1.89·5-s − 1.66i·13-s − 1.71i·17-s + 2.59·25-s − 0.787·29-s + 1.97i·37-s − 0.220i·41-s + 49-s − 1.74·53-s + 1.53i·61-s − 3.15i·65-s + 1.87·73-s − 3.25i·85-s + 1.94i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269091941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269091941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 12iT - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18.3iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.727380091279822037469095724260, −9.159711438347431716558426262062, −8.132885249342203097285146732588, −7.12200634456221861534024840262, −6.23345177835311418834626686350, −5.43353300245555024798488117957, −4.92632111829821756938065375860, −3.14220198257125770351731104344, −2.41563130902469822184928660742, −1.06128194526495515645448028093,
1.66790207971258112415890313904, 2.17109323394852428493266134422, 3.72521529742037320006668361241, 4.82224861148872943004626448403, 5.88555992163844546460120828132, 6.29559020225044511567527529003, 7.20558287386730506340138791691, 8.495889680336292458082750154009, 9.264585457094972309932373698447, 9.687428625045475015153544999526
