L(s) = 1 | + (−1.40 + 0.199i)2-s + (0.520 + 1.65i)3-s + (1.92 − 0.557i)4-s + (−1.05 − 2.20i)6-s − 1.92i·7-s + (−2.57 + 1.16i)8-s + (−2.45 + 1.72i)9-s − 4.02i·11-s + (1.92 + 2.88i)12-s − 4.81i·13-s + (0.383 + 2.69i)14-s + (3.37 − 2.14i)16-s − 5.23i·17-s + (3.09 − 2.89i)18-s − 0.684·19-s + ⋯ |
L(s) = 1 | + (−0.990 + 0.140i)2-s + (0.300 + 0.953i)3-s + (0.960 − 0.278i)4-s + (−0.431 − 0.901i)6-s − 0.728i·7-s + (−0.911 + 0.411i)8-s + (−0.819 + 0.573i)9-s − 1.21i·11-s + (0.554 + 0.832i)12-s − 1.33i·13-s + (0.102 + 0.721i)14-s + (0.844 − 0.535i)16-s − 1.26i·17-s + (0.730 − 0.682i)18-s − 0.157·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.743531 - 0.332865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.743531 - 0.332865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.199i)T \) |
| 3 | \( 1 + (-0.520 - 1.65i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.92iT - 7T^{2} \) |
| 11 | \( 1 + 4.02iT - 11T^{2} \) |
| 13 | \( 1 + 4.81iT - 13T^{2} \) |
| 17 | \( 1 + 5.23iT - 17T^{2} \) |
| 19 | \( 1 + 0.684T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 + 6.99T + 29T^{2} \) |
| 31 | \( 1 + 4.23iT - 31T^{2} \) |
| 37 | \( 1 - 9.83iT - 37T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 1.04T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 0.994iT - 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 9.25iT - 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 + 0.829iT - 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−10.44953296210980711918184397510, −9.681523514738081237044097311197, −8.872059360324746589783592351147, −8.063201589330432046985507388050, −7.31760038042138142818357666837, −5.97810463541579005044003479882, −5.14589474179993502588136061158, −3.59703252686896875065473086952, −2.72455350271387836983641896944, −0.59064582887046735324888415993,
1.68364979923723920309077801284, 2.32551710027869157371476561360, 3.86977153590700732537070675020, 5.66468925688234053475584326331, 6.66817257914868271424433877144, 7.26849174215393926239126793592, 8.234202682317061845384245205535, 9.037798388652386080671216045641, 9.574682226821286951139804604260, 10.83605856475605451472500804550