| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.348 − 0.155i)3-s + (0.309 + 0.951i)4-s + (0.413 − 0.716i)5-s + (−0.190 − 0.330i)6-s + (−0.981 + 1.09i)7-s + (−0.309 + 0.951i)8-s + (−1.90 − 2.12i)9-s + (0.755 − 0.336i)10-s + (−2.16 − 0.459i)11-s + (0.0399 − 0.379i)12-s + (−0.199 − 1.90i)13-s + (−1.43 + 0.305i)14-s + (−0.255 + 0.185i)15-s + (−0.809 + 0.587i)16-s + (3.17 − 0.674i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.201 − 0.0896i)3-s + (0.154 + 0.475i)4-s + (0.184 − 0.320i)5-s + (−0.0779 − 0.135i)6-s + (−0.371 + 0.412i)7-s + (−0.109 + 0.336i)8-s + (−0.636 − 0.707i)9-s + (0.238 − 0.106i)10-s + (−0.651 − 0.138i)11-s + (0.0115 − 0.109i)12-s + (−0.0554 − 0.527i)13-s + (−0.383 + 0.0815i)14-s + (−0.0659 + 0.0479i)15-s + (−0.202 + 0.146i)16-s + (0.769 − 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02771 + 0.227616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02771 + 0.227616i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.201 - 5.56i)T \) |
| good | 3 | \( 1 + (0.348 + 0.155i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.413 + 0.716i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.981 - 1.09i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (2.16 + 0.459i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.199 + 1.90i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-3.17 + 0.674i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.657 - 6.26i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.37 + 4.22i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.781 + 0.567i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-1.34 - 2.32i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.59 + 4.27i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.06 + 10.1i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (4.19 - 3.04i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (8.45 + 9.38i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (2.95 + 1.31i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + (-0.100 + 0.174i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.66 + 8.51i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-2.44 - 0.519i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-12.4 + 2.63i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.08 + 0.927i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-5.29 - 16.2i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.28 + 7.02i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−15.01103759514426432476562865335, −14.12114568173750706871125632209, −12.71151648501904553325019865341, −12.21531165433793870855565886279, −10.65794140242930362077575821475, −9.138688425471260223598883590657, −7.87299876478642353748652260836, −6.23124071506102780257207945864, −5.28566759733401299641919198799, −3.22923032353409435971154435935,
2.78168076086641373317387597992, 4.69821221689604822434359648625, 6.10113108237928422046358807716, 7.61498844561070515061111980230, 9.448014156486140778063169735536, 10.66201362801519652512392395767, 11.44432028632346229463276924912, 12.88863843768137717470246907179, 13.72825149677688589977821143733, 14.72692753843056951866177412580
