L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.72 − 0.128i)3-s − 1.00i·4-s + (−1.56 + 1.60i)5-s + (1.31 − 1.13i)6-s + (−0.236 − 0.236i)7-s + (0.707 + 0.707i)8-s + (2.96 + 0.445i)9-s + (−0.0281 − 2.23i)10-s − 2.01·11-s + (−0.128 + 1.72i)12-s + (−3.47 − 0.965i)13-s + 0.334·14-s + (2.90 − 2.56i)15-s − 1.00·16-s + (−2.69 − 2.69i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.997 − 0.0743i)3-s − 0.500i·4-s + (−0.698 + 0.715i)5-s + (0.535 − 0.461i)6-s + (−0.0893 − 0.0893i)7-s + (0.250 + 0.250i)8-s + (0.988 + 0.148i)9-s + (−0.00888 − 0.707i)10-s − 0.606·11-s + (−0.0371 + 0.498i)12-s + (−0.963 − 0.267i)13-s + 0.0893·14-s + (0.749 − 0.662i)15-s − 0.250·16-s + (−0.652 − 0.652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.387291 - 0.183896i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.387291 - 0.183896i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.72 + 0.128i)T \) |
| 5 | \( 1 + (1.56 - 1.60i)T \) |
| 13 | \( 1 + (3.47 + 0.965i)T \) |
good | 7 | \( 1 + (0.236 + 0.236i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 17 | \( 1 + (2.69 + 2.69i)T + 17iT^{2} \) |
| 19 | \( 1 - 8.01T + 19T^{2} \) |
| 23 | \( 1 + (-6.23 + 6.23i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 + 7.05iT - 31T^{2} \) |
| 37 | \( 1 + (1.19 + 1.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.90T + 41T^{2} \) |
| 43 | \( 1 + (-2.76 - 2.76i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.04 + 2.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.83 - 6.83i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + (-4.26 - 4.26i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.88T + 71T^{2} \) |
| 73 | \( 1 + (2.59 - 2.59i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 + (2.98 + 2.98i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.02iT - 89T^{2} \) |
| 97 | \( 1 + (10.9 + 10.9i)T + 97iT^{2} \) |
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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
−11.15140796851323185245344480669, −10.29729967742973444514824843840, −9.576296341161241859407727066273, −8.126355950008814436937005592331, −7.20688765244502524165191892201, −6.77284670830589416201125683449, −5.42171577756277967293542765003, −4.58646602608554490367607387862, −2.79469435024722999005706065912, −0.42996288439946438894194342627,
1.26161394024179732240234973397, 3.27713576568980162281842916403, 4.69356658784969155525685996214, 5.35871686148053001061852769519, 6.98702537242194631615739438774, 7.65369074542363062434620855747, 8.882913140159821500923430551956, 9.713655342130819948317509160507, 10.62936968022657485360098535077, 11.53617643967983869854207270597