L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (1.84 + 1.26i)5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.410 − 2.19i)10-s + 5.16·11-s + (−0.707 + 0.707i)12-s + (−0.184 − 0.184i)13-s + (0.410 + 2.19i)15-s − 1.00·16-s + (0.750 − 0.750i)17-s + (0.707 − 0.707i)18-s + 4.08·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.824 + 0.565i)5-s − 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.129 − 0.695i)10-s + 1.55·11-s + (−0.204 + 0.204i)12-s + (−0.0512 − 0.0512i)13-s + (0.106 + 0.567i)15-s − 0.250·16-s + (0.182 − 0.182i)17-s + (0.166 − 0.166i)18-s + 0.937·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960907219\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960907219\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.84 - 1.26i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + (0.184 + 0.184i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.750 + 0.750i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 23 | \( 1 + (2.36 - 2.36i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.387iT - 29T^{2} \) |
| 31 | \( 1 + 10.6iT - 31T^{2} \) |
| 37 | \( 1 + (-2.51 - 2.51i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.05iT - 41T^{2} \) |
| 43 | \( 1 + (2.13 - 2.13i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.37 - 7.37i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.15 + 7.15i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.70T + 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 + (0.0355 + 0.0355i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + (-3.22 - 3.22i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.46iT - 79T^{2} \) |
| 83 | \( 1 + (-0.409 - 0.409i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + (1.32 - 1.32i)T - 97iT^{2} \) |
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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
−9.645120128960034884206416794294, −9.113076581806004904874620167394, −8.092230508794671629645193152253, −7.23129457897842087174500593898, −6.37285231659484692678966164415, −5.46130300092162467879099405659, −4.16619967559218030471683576775, −3.38936716403777943822727951241, −2.38020569454196036848873740058, −1.31709004389374278911818663096,
1.07379590188401452319889477509, 1.87277796171372457621078023635, 3.31257999946253393164826675208, 4.54486459067485056532241918655, 5.51375787648337863069534242414, 6.43729645235570031662164534307, 6.89315532534619891011504002028, 8.025502222903773197314942774909, 8.674698992637338541335599552676, 9.377288774477209778661461839491