| L(s) = 1 | + (0.891 + 0.453i)2-s + (1.73 + 0.00452i)3-s + (0.587 + 0.809i)4-s + (−1.35 − 1.78i)5-s + (1.54 + 0.790i)6-s + (−0.152 − 0.152i)7-s + (0.156 + 0.987i)8-s + (2.99 + 0.0156i)9-s + (−0.396 − 2.20i)10-s + (−4.88 + 1.58i)11-s + (1.01 + 1.40i)12-s + (0.674 + 1.32i)13-s + (−0.0667 − 0.205i)14-s + (−2.33 − 3.09i)15-s + (−0.309 + 0.951i)16-s + (−4.81 + 0.762i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.999 + 0.00261i)3-s + (0.293 + 0.404i)4-s + (−0.604 − 0.796i)5-s + (0.629 + 0.322i)6-s + (−0.0577 − 0.0577i)7-s + (0.0553 + 0.349i)8-s + (0.999 + 0.00522i)9-s + (−0.125 − 0.695i)10-s + (−1.47 + 0.478i)11-s + (0.292 + 0.405i)12-s + (0.187 + 0.367i)13-s + (−0.0178 − 0.0548i)14-s + (−0.602 − 0.797i)15-s + (−0.0772 + 0.237i)16-s + (−1.16 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77939 + 0.232641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77939 + 0.232641i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 3 | \( 1 + (-1.73 - 0.00452i)T \) |
| 5 | \( 1 + (1.35 + 1.78i)T \) |
| good | 7 | \( 1 + (0.152 + 0.152i)T + 7iT^{2} \) |
| 11 | \( 1 + (4.88 - 1.58i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.674 - 1.32i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (4.81 - 0.762i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.283 + 0.389i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.21 + 2.38i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-7.59 + 5.51i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.84 + 1.33i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 1.95i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (5.95 + 1.93i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.72 - 2.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.58 - 10.0i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-7.59 - 1.20i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.54 + 4.74i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.21 + 12.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.35 - 14.8i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-7.13 - 9.81i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.36 - 4.26i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (1.28 + 1.76i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.782 - 4.93i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.38 - 10.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (9.96 + 1.57i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.09588933885181047313191408780, −12.54078903634422811719805405637, −11.19987165301886422656613406435, −9.864207971028380200328724051147, −8.573791393931669499558904867279, −7.934766307291803038700894839983, −6.79094173801939817317490165114, −4.97571894904420650394518730577, −4.09204956617238406811800775362, −2.48546535659345332082912386210,
2.57545352703517314412054134885, 3.46610289027785041779072439014, 4.93089777875704841098257995158, 6.63937461018284939248321795028, 7.70959211186331166759870224495, 8.715290972972413037205994455161, 10.25475164173442603596590505291, 10.84550887969271815098256379751, 12.10620443059829214263638787563, 13.27858056769714210449765196842
