L(s) = 1 | + (−2.94 + 0.581i)3-s + 11.4i·7-s + (8.32 − 3.42i)9-s − 8.48i·11-s − 10i·13-s − 3.55·17-s − 10.9·19-s + (−6.67 − 33.8i)21-s − 17.6·23-s + (−22.5 + 14.9i)27-s − 26.8i·29-s − 8·31-s + (4.93 + 24.9i)33-s + 59.9i·37-s + (5.81 + 29.4i)39-s + ⋯ |
L(s) = 1 | + (−0.981 + 0.193i)3-s + 1.64i·7-s + (0.924 − 0.380i)9-s − 0.771i·11-s − 0.769i·13-s − 0.209·17-s − 0.577·19-s + (−0.317 − 1.60i)21-s − 0.767·23-s + (−0.833 + 0.552i)27-s − 0.925i·29-s − 0.258·31-s + (0.149 + 0.756i)33-s + 1.62i·37-s + (0.149 + 0.754i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8643555807\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8643555807\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.94 - 0.581i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.4iT - 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 + 3.55T + 289T^{2} \) |
| 19 | \( 1 + 10.9T + 361T^{2} \) |
| 23 | \( 1 + 17.6T + 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 59.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 88.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 3.55T + 2.80e3T^{2} \) |
| 59 | \( 1 + 77.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 21.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 9.02T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.688T + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.10iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111. iT - 9.40e3T^{2} \) |
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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
−9.498017786724275522190236093074, −8.634931246503768965610946087434, −7.944385652467487497163180895900, −6.62788876203397693361560956578, −5.89053533563478915492739729082, −5.46048502337782822946730447023, −4.41391403240051678862595289507, −3.16976294585930221819404727759, −1.99943413316431309783837867279, −0.36641356952153239996427159973,
0.933578979125606430085560115658, 2.06488933844797202257250314781, 4.02638322279851492857298008523, 4.30326856700036909941712595859, 5.45564580843613045226101976790, 6.50971840141920402070110917742, 7.16915795870692040188604446210, 7.63104447637744610990521583669, 8.977008474627605171611200355998, 9.938182066663411213346705511359