L(s) = 1 | + 4.47i·2-s + (0.787 − 5.13i)3-s − 12.0·4-s − 5·5-s + (22.9 + 3.52i)6-s + (−18.2 + 3.09i)7-s − 18.1i·8-s + (−25.7 − 8.08i)9-s − 22.3i·10-s − 29.0i·11-s + (−9.48 + 61.8i)12-s − 71.4i·13-s + (−13.8 − 81.7i)14-s + (−3.93 + 25.6i)15-s − 15.2·16-s − 55.3·17-s + ⋯ |
L(s) = 1 | + 1.58i·2-s + (0.151 − 0.988i)3-s − 1.50·4-s − 0.447·5-s + (1.56 + 0.239i)6-s + (−0.985 + 0.167i)7-s − 0.801i·8-s + (−0.954 − 0.299i)9-s − 0.707i·10-s − 0.794i·11-s + (−0.228 + 1.48i)12-s − 1.52i·13-s + (−0.264 − 1.56i)14-s + (−0.0677 + 0.442i)15-s − 0.237·16-s − 0.789·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.232905 - 0.229207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.232905 - 0.229207i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.787 + 5.13i)T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + (18.2 - 3.09i)T \) |
good | 2 | \( 1 - 4.47iT - 8T^{2} \) |
| 11 | \( 1 + 29.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 71.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 55.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.58iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 80.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 177. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 66.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 70.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 504. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 392.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 724. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 341. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 689. iT - 9.12e5T^{2} \) |
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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
−13.26092565729044990539365059537, −12.45902930574607516409044543872, −10.96549889614994463895143199631, −9.138103018342896382770909159067, −8.239953038639727470326346174952, −7.33949426219544346621784302388, −6.34759457349510092029941212156, −5.45315400942630855295645748695, −3.21694312525677718221206723681, −0.17007384550787670741223894083,
2.38784882806862909196506275531, 3.80390551442493144962965321160, 4.55762872237526916086464131881, 6.74892880244033684070587430865, 8.799504759653718265175410728418, 9.575317134272400257336296102899, 10.37891872376534538606286569786, 11.38126744632217614253524133505, 12.18584089878910007243992176661, 13.29053680413090497645981638145