L(s) = 1 | + (−1.85 − 1.85i)2-s + (−0.170 − 1.72i)3-s + 4.85i·4-s + (−2.22 + 0.265i)5-s + (−2.87 + 3.50i)6-s + (5.29 − 5.29i)8-s + (−2.94 + 0.588i)9-s + (4.60 + 3.62i)10-s − 2.17i·11-s + (8.37 − 0.829i)12-s + (2.87 + 2.87i)13-s + (0.835 + 3.78i)15-s − 9.89·16-s + (4.03 + 4.03i)17-s + (6.53 + 4.35i)18-s + 3.45i·19-s + ⋯ |
L(s) = 1 | + (−1.30 − 1.30i)2-s + (−0.0985 − 0.995i)3-s + 2.42i·4-s + (−0.992 + 0.118i)5-s + (−1.17 + 1.43i)6-s + (1.87 − 1.87i)8-s + (−0.980 + 0.196i)9-s + (1.45 + 1.14i)10-s − 0.657i·11-s + (2.41 − 0.239i)12-s + (0.797 + 0.797i)13-s + (0.215 + 0.976i)15-s − 2.47·16-s + (0.979 + 0.979i)17-s + (1.54 + 1.02i)18-s + 0.792i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.321733 - 0.453181i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.321733 - 0.453181i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.170 + 1.72i)T \) |
| 5 | \( 1 + (2.22 - 0.265i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.85 + 1.85i)T + 2iT^{2} \) |
| 11 | \( 1 + 2.17iT - 11T^{2} \) |
| 13 | \( 1 + (-2.87 - 2.87i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.03 - 4.03i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.45iT - 19T^{2} \) |
| 23 | \( 1 + (-1.26 + 1.26i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + (-6.85 + 6.85i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.38iT - 41T^{2} \) |
| 43 | \( 1 + (1.85 + 1.85i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.02 - 1.02i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.260 + 0.260i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.25T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 + (-6.54 + 6.54i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.5iT - 71T^{2} \) |
| 73 | \( 1 + (1.31 + 1.31i)T + 73iT^{2} \) |
| 79 | \( 1 - 6.89iT - 79T^{2} \) |
| 83 | \( 1 + (2.04 - 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (-12.4 + 12.4i)T - 97iT^{2} \) |
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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
−10.35911725194935482490339700857, −9.114085184954200358600440724487, −8.429847548216810198244925704131, −7.897469411387226249291352781876, −7.10100076004562605097678672055, −5.92605451809714738311170966540, −3.93656503854638235867459468614, −3.23831697355137081441342432264, −1.89052504372559980447926500236, −0.78686341599622144128281930012,
0.75294067208519589007860442218, 3.21925782717097682128880356563, 4.66283272802256018842440828575, 5.35904338827691729630109028879, 6.42582453358227570092900481763, 7.47141403578612909512165274927, 8.038383643780675780562673665019, 8.908815040838636851256837826962, 9.541743211662051190543504289626, 10.33364887834015945453793491212