L(s) = 1 | + (−2.12 + 0.707i)5-s − 6i·13-s − 4.24·17-s + (3.99 − 3i)25-s − 9.89i·29-s − 12i·37-s + 1.41i·41-s − 7·49-s − 12.7·53-s + 10·61-s + (4.24 + 12.7i)65-s + 6i·73-s + (8.99 − 3i)85-s − 18.3i·89-s + 18i·97-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.316i)5-s − 1.66i·13-s − 1.02·17-s + (0.799 − 0.600i)25-s − 1.83i·29-s − 1.97i·37-s + 0.220i·41-s − 49-s − 1.74·53-s + 1.28·61-s + (0.526 + 1.57i)65-s + 0.702i·73-s + (0.976 − 0.325i)85-s − 1.94i·89-s + 1.82i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445503 - 0.600193i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445503 - 0.600193i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.12 - 0.707i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 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 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.31103464991963411185556447516, −9.291539929188379949268398126605, −8.178872267354683492453520209175, −7.74716618433331204421098351241, −6.69667965364816107529369637403, −5.69712165555834243556353144971, −4.54281210948764864472564607902, −3.57596980410455058267894633976, −2.48772372822354316534753994753, −0.38860322550039365000306323842,
1.62797325025977835922967659681, 3.22155136065658197588202173122, 4.33197709927822494159145214266, 4.96780954512086566171453103480, 6.53507383978101453033125103184, 7.04998894074605768132287672407, 8.218196826637545485422098828944, 8.867355243519122434472487534009, 9.657308378933955520645306362935, 10.92655953516176587191793330710