L(s) = 1 | + 2-s + (1.03 + 1.38i)3-s + 4-s + (1.98 − 1.02i)5-s + (1.03 + 1.38i)6-s + 8-s + (−0.844 + 2.87i)9-s + (1.98 − 1.02i)10-s + 1.27i·11-s + (1.03 + 1.38i)12-s − 1.32·13-s + (3.48 + 1.69i)15-s + 16-s + 5.76i·17-s + (−0.844 + 2.87i)18-s + 2.44i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.599 + 0.800i)3-s + 0.5·4-s + (0.888 − 0.457i)5-s + (0.423 + 0.566i)6-s + 0.353·8-s + (−0.281 + 0.959i)9-s + (0.628 − 0.323i)10-s + 0.385i·11-s + (0.299 + 0.400i)12-s − 0.366·13-s + (0.899 + 0.437i)15-s + 0.250·16-s + 1.39i·17-s + (−0.199 + 0.678i)18-s + 0.560i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.694865593\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.694865593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.03 - 1.38i)T \) |
| 5 | \( 1 + (-1.98 + 1.02i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.27iT - 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 0.680T + 23T^{2} \) |
| 29 | \( 1 + 6.12iT - 29T^{2} \) |
| 31 | \( 1 - 6.94iT - 31T^{2} \) |
| 37 | \( 1 + 4.67iT - 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 1.87iT - 43T^{2} \) |
| 47 | \( 1 + 7.97iT - 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 4.22iT - 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 5.09iT - 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.33iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−9.765331383039719540090012232387, −8.860097896508466422670263126378, −8.177796028992322510794162340687, −7.17440665748041890382242932174, −6.01182731580348744876353719930, −5.43608492498662276297132574015, −4.48241975418202034503154590386, −3.81594774737110351218460931717, −2.59583309117432423427292621054, −1.75292277326586254658789454548,
1.17803598760064356790883579837, 2.62255632999716008396975828532, 2.82959605243097739100686745265, 4.24021644299553464862509267723, 5.42613444806551396233124239564, 6.07737714604361103356289858428, 7.07804232596090467464191285998, 7.35529724068502888827232818522, 8.606767888933761344792344853943, 9.363395177594366045353745052193