L(s) = 1 | + (1.58 − 1.58i)2-s − 3.00i·4-s + (−2 − i)5-s + (−0.581 − 2.58i)7-s + (−1.58 − 1.58i)8-s + (−4.74 + 1.58i)10-s − 3.16·11-s + (3.16 − 3.16i)13-s + (−5 − 3.16i)14-s + 0.999·16-s + (5 + 5i)17-s + 3.16·19-s + (−3.00 + 6.00i)20-s + (−5.00 + 5.00i)22-s + (−3.16 − 3.16i)23-s + ⋯ |
L(s) = 1 | + (1.11 − 1.11i)2-s − 1.50i·4-s + (−0.894 − 0.447i)5-s + (−0.219 − 0.975i)7-s + (−0.559 − 0.559i)8-s + (−1.50 + 0.500i)10-s − 0.953·11-s + (0.877 − 0.877i)13-s + (−1.33 − 0.845i)14-s + 0.249·16-s + (1.21 + 1.21i)17-s + 0.725·19-s + (−0.670 + 1.34i)20-s + (−1.06 + 1.06i)22-s + (−0.659 − 0.659i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.733925 - 1.79816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.733925 - 1.79816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (0.581 + 2.58i)T \) |
good | 2 | \( 1 + (-1.58 + 1.58i)T - 2iT^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + (3.16 + 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.16iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 + (9.48 - 9.48i)T - 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + (10 - 10i)T - 83iT^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \) |
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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
−11.37957216907948087319172002323, −10.59604132352909514701341423847, −10.07179576320685262510351943550, −8.279869939177094373551300680687, −7.64045651347113693533527481797, −5.94275622591943317077720891750, −4.90813605028666147660141490915, −3.83011851363012797057816020424, −3.16872764011861425160410172663, −1.12435155721763894921433355657,
2.93917166664466473891200142163, 3.95685391782094549281882096905, 5.24933649302041899440759733710, 5.94775354160537150406710101893, 7.17896517752495296524741139027, 7.74789889305121491726985693620, 8.863735934703529925946511368597, 10.17049613438773577043111023182, 11.62191499902165436421703093144, 12.01306007893170743511877249431