L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s + (1.67 + 1.48i)5-s − 1.73·6-s + (2 − 2i)7-s + (0.707 − 0.707i)8-s − 2.99i·9-s + (−0.133 − 2.23i)10-s − 2.31i·11-s + (1.22 + 1.22i)12-s + (3.36 + 3.36i)13-s − 2.82·14-s + (3.86 − 0.232i)15-s − 1.00·16-s + (−0.707 − 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.707 − 0.707i)3-s + 0.500i·4-s + (0.748 + 0.663i)5-s − 0.707·6-s + (0.755 − 0.755i)7-s + (0.250 − 0.250i)8-s − 0.999i·9-s + (−0.0423 − 0.705i)10-s − 0.696i·11-s + (0.353 + 0.353i)12-s + (0.933 + 0.933i)13-s − 0.755·14-s + (0.998 − 0.0599i)15-s − 0.250·16-s + (−0.171 − 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58721 - 1.18054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58721 - 1.18054i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.31iT - 11T^{2} \) |
| 13 | \( 1 + (-3.36 - 3.36i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + 17iT^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 37 | \( 1 + (0.464 - 0.464i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-3.46 - 3.46i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.189 - 0.189i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.89 - 4.89i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + (-0.169 + 0.169i)T - 67iT^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + (9.46 + 9.46i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.73iT - 79T^{2} \) |
| 83 | \( 1 + (4.43 - 4.43i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + (8.36 - 8.36i)T - 97iT^{2} \) |
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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
−9.810719322616079483460295573999, −9.023896574271439578266607618873, −8.340828891248057473784715476948, −7.43398011297721304321587641552, −6.73627062811984027178115679687, −5.81376017620633419163026734156, −4.14975675347117337125165469115, −3.25654876085376603150565709073, −2.07947240219472408664006697898, −1.21354039676173014338116008607,
1.53794715201500429327008880443, 2.60110090098288527662325285209, 4.14084141125305610164538372945, 5.25800839134963510428848778882, 5.57583282502306350281295510506, 6.99176791643592202353709559694, 8.138309154734410307495692150739, 8.559718968362633775860130277272, 9.295618855461325293476775572418, 9.918844242847229885783329926381