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
−9.377288774477209778661461839491, −8.674698992637338541335599552676, −8.025502222903773197314942774909, −6.89315532534619891011504002028, −6.43729645235570031662164534307, −5.51375787648337863069534242414, −4.54486459067485056532241918655, −3.31257999946253393164826675208, −1.87277796171372457621078023635, −1.07379590188401452319889477509,
1.31709004389374278911818663096, 2.38020569454196036848873740058, 3.38936716403777943822727951241, 4.16619967559218030471683576775, 5.46130300092162467879099405659, 6.37285231659484692678966164415, 7.23129457897842087174500593898, 8.092230508794671629645193152253, 9.113076581806004904874620167394, 9.645120128960034884206416794294