| L(s) = 1 | + (−1.39 + 0.235i)2-s + (−1.04 − 1.38i)3-s + (1.88 − 0.657i)4-s + (1.89 − 1.18i)5-s + (1.77 + 1.68i)6-s + 4.72i·7-s + (−2.47 + 1.36i)8-s + (−0.830 + 2.88i)9-s + (−2.36 + 2.10i)10-s + (2.39 − 1.73i)11-s + (−2.87 − 1.93i)12-s + (3.16 + 2.29i)13-s + (−1.11 − 6.59i)14-s + (−3.61 − 1.38i)15-s + (3.13 − 2.48i)16-s + (−0.528 + 0.171i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 + 0.166i)2-s + (−0.601 − 0.799i)3-s + (0.944 − 0.328i)4-s + (0.847 − 0.530i)5-s + (0.725 + 0.687i)6-s + 1.78i·7-s + (−0.876 + 0.481i)8-s + (−0.276 + 0.960i)9-s + (−0.747 + 0.664i)10-s + (0.721 − 0.524i)11-s + (−0.830 − 0.557i)12-s + (0.877 + 0.637i)13-s + (−0.297 − 1.76i)14-s + (−0.933 − 0.358i)15-s + (0.784 − 0.620i)16-s + (−0.128 + 0.0416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.843906 + 0.0105792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.843906 + 0.0105792i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.235i)T \) |
| 3 | \( 1 + (1.04 + 1.38i)T \) |
| 5 | \( 1 + (-1.89 + 1.18i)T \) |
| good | 7 | \( 1 - 4.72iT - 7T^{2} \) |
| 11 | \( 1 + (-2.39 + 1.73i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.16 - 2.29i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.528 - 0.171i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.14 - 0.373i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.664 - 0.482i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.07 - 1.97i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.42 + 1.43i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.51 - 2.55i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.90 - 2.62i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.83iT - 43T^{2} \) |
| 47 | \( 1 + (-1.36 + 4.20i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.98 + 2.59i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.804 + 0.584i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.40 - 3.20i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.581 - 0.188i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.78 - 14.7i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.18 + 5.94i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-14.2 - 4.64i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.119 - 0.366i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.82 + 8.01i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.82 - 14.8i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.84723374658457246841967660765, −10.88046005376625391954316814074, −9.634388334509541046151206939367, −8.707938364957130762809444718699, −8.340005843970882464966081395769, −6.52907930089252595719096458979, −6.16892543703676221666975666487, −5.23695371957631686001533745002, −2.49247395763535584501389300519, −1.41112860593219500696982451841,
1.11555719586362169768429168527, 3.23534176053141554737866484262, 4.42309884542821555311320476038, 6.21783730128006231013696799136, 6.72182133986209434880035096897, 7.955405642725428020907569576494, 9.332519512899903353138007668017, 9.993114559637749126489889947685, 10.70306426694291148942706719884, 11.10601946521701333673940467221
