L(s) = 1 | + (−0.627 − 1.26i)2-s + (−1.21 + 1.59i)4-s + (−2.15 + 1.24i)5-s + (2.64 + 0.0803i)7-s + (2.77 + 0.537i)8-s + (2.92 + 1.95i)10-s + (2.30 − 3.99i)11-s + 5.22·13-s + (−1.55 − 3.40i)14-s + (−1.06 − 3.85i)16-s + (4.85 + 2.80i)17-s + (−2.76 + 1.59i)19-s + (0.632 − 4.93i)20-s + (−6.50 − 0.414i)22-s + (0.359 + 0.622i)23-s + ⋯ |
L(s) = 1 | + (−0.443 − 0.896i)2-s + (−0.605 + 0.795i)4-s + (−0.963 + 0.556i)5-s + (0.999 + 0.0303i)7-s + (0.981 + 0.189i)8-s + (0.926 + 0.616i)10-s + (0.694 − 1.20i)11-s + 1.44·13-s + (−0.416 − 0.909i)14-s + (−0.265 − 0.964i)16-s + (1.17 + 0.680i)17-s + (−0.634 + 0.366i)19-s + (0.141 − 1.10i)20-s + (−1.38 − 0.0884i)22-s + (0.0749 + 0.129i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.924404 - 0.312952i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.924404 - 0.312952i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.627 + 1.26i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.64 - 0.0803i)T \) |
good | 5 | \( 1 + (2.15 - 1.24i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.30 + 3.99i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 + (-4.85 - 2.80i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.76 - 1.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.359 - 0.622i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.53iT - 29T^{2} \) |
| 31 | \( 1 + (-1.01 - 0.588i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.35 + 2.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.83iT - 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + (-2.70 - 4.69i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.79 + 1.03i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.05 - 3.56i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.505 - 0.874i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.9 + 6.32i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + (4.81 - 8.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.65 - 4.41i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + (-7.38 + 4.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−11.65958545379126840608556790539, −11.01956678547530956360369942384, −10.48143234614359246119364654679, −8.778969159447296917050539057246, −8.362207449717756261353112752149, −7.35206056640432557467647005676, −5.76596565729353103384047408127, −4.01359345017091917501052249275, −3.38861694933662599109919890934, −1.35026806530563768339945420621,
1.27245732916936603556161262726, 4.11832468676270440946034829190, 4.84214174587577937106823444987, 6.22709123032550284094644069515, 7.45620259342788297767510898261, 8.137077208450301050172142381077, 8.937737837263563752362102228270, 10.03338702164257018609197609043, 11.24100414250239434503968281062, 12.00095816486724235350469369485