| L(s) = 1 | + (0.0527 − 0.0726i)2-s + (0.951 − 0.309i)3-s + (0.615 + 1.89i)4-s + (−1.27 + 1.83i)5-s + (0.0277 − 0.0854i)6-s − 4.36i·7-s + (0.341 + 0.110i)8-s + (0.809 − 0.587i)9-s + (0.0665 + 0.189i)10-s + (−3.55 − 2.58i)11-s + (1.17 + 1.61i)12-s + (1.16 + 1.60i)13-s + (−0.316 − 0.230i)14-s + (−0.640 + 2.14i)15-s + (−3.19 + 2.32i)16-s + (−0.948 − 0.308i)17-s + ⋯ |
| L(s) = 1 | + (0.0373 − 0.0513i)2-s + (0.549 − 0.178i)3-s + (0.307 + 0.947i)4-s + (−0.568 + 0.822i)5-s + (0.0113 − 0.0348i)6-s − 1.64i·7-s + (0.120 + 0.0391i)8-s + (0.269 − 0.195i)9-s + (0.0210 + 0.0599i)10-s + (−1.07 − 0.778i)11-s + (0.337 + 0.465i)12-s + (0.323 + 0.444i)13-s + (−0.0846 − 0.0615i)14-s + (−0.165 + 0.553i)15-s + (−0.799 + 0.580i)16-s + (−0.229 − 0.0747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04435 + 0.123660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04435 + 0.123660i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 + (1.27 - 1.83i)T \) |
| good | 2 | \( 1 + (-0.0527 + 0.0726i)T + (-0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + 4.36iT - 7T^{2} \) |
| 11 | \( 1 + (3.55 + 2.58i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 1.60i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.948 + 0.308i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.417 - 1.28i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.38 + 1.90i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 7.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.13 - 3.49i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.844 + 1.16i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.83 + 3.51i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.68iT - 43T^{2} \) |
| 47 | \( 1 + (-10.4 + 3.38i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (10.5 - 3.41i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.41 + 3.93i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.64 - 5.55i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (12.2 + 3.99i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.26 - 6.96i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.249 - 0.343i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.96 - 6.04i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.700 - 0.227i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (7.91 + 5.75i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0320 + 0.0104i)T + (78.4 - 57.0i)T^{2} \) |
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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
−14.32280355828488644073289657895, −13.60394321831523776995868145249, −12.56998541777538364143222149209, −11.07935706120095253850122036864, −10.52926599123949024101065197101, −8.528320654217548530322397103705, −7.53668519408803716183303292290, −6.83250719885505848261988936531, −4.08824049303585841750350566965, −3.03306794474273471313672482081,
2.38510189573010561608721393985, 4.79063880517486030353404733894, 5.83216752785096699352755528395, 7.75415059260441266254417884751, 8.891628938844952042425091248769, 9.833309469217743819875269658940, 11.25606868255527509713622770346, 12.39800013970698941893053009440, 13.37502277836905565860324872318, 14.97371990856422705431903024546
