| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.0399 − 0.379i)3-s + (0.309 − 0.951i)4-s + (−0.604 + 1.04i)5-s + (−0.190 − 0.330i)6-s + (−3.01 + 0.641i)7-s + (−0.309 − 0.951i)8-s + (2.79 + 0.593i)9-s + (0.126 + 1.20i)10-s + (0.115 + 0.128i)11-s + (−0.348 − 0.155i)12-s + (−4.27 + 1.90i)13-s + (−2.06 + 2.29i)14-s + (0.373 + 0.271i)15-s + (−0.809 − 0.587i)16-s + (1.91 − 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.0230 − 0.219i)3-s + (0.154 − 0.475i)4-s + (−0.270 + 0.468i)5-s + (−0.0779 − 0.135i)6-s + (−1.14 + 0.242i)7-s + (−0.109 − 0.336i)8-s + (0.930 + 0.197i)9-s + (0.0399 + 0.380i)10-s + (0.0348 + 0.0387i)11-s + (−0.100 − 0.0448i)12-s + (−1.18 + 0.527i)13-s + (−0.551 + 0.612i)14-s + (0.0964 + 0.0700i)15-s + (−0.202 − 0.146i)16-s + (0.464 − 0.516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02521 - 0.294313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02521 - 0.294313i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.10 + 4.61i)T \) |
| good | 3 | \( 1 + (-0.0399 + 0.379i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (0.604 - 1.04i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3.01 - 0.641i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.115 - 0.128i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (4.27 - 1.90i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.91 + 2.12i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.269 + 0.119i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.53 + 4.72i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.82 + 4.95i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-4.00 - 6.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.196 + 1.86i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (4.41 + 1.96i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-8.11 - 5.89i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (8.21 + 1.74i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (1.54 - 14.7i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + (4.60 - 7.96i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.97 - 2.12i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-1.57 - 1.74i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-4.94 + 5.48i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.582 + 5.54i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.590 + 1.81i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.32 + 16.3i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.84939295967019173366693090747, −13.67742671941804360067773076546, −12.63293972765083559057864430573, −11.87151083628377754440094164763, −10.33684089293989537742877257571, −9.502087109237385278241616289622, −7.41515718185044930684651710954, −6.36505637056999274971184602546, −4.48597119978132507112015533799, −2.75713476615165345604856051944,
3.48450706379823773720029534292, 4.93257207526502555921603186419, 6.56605909336514256392344611365, 7.74364902116984926523943020129, 9.389812954512332648680422979872, 10.43475887393154285250974717499, 12.43015073959315422389143809846, 12.64232102116782188424085186491, 14.05904638375445312309004916903, 15.27255544747811180341670252248
