L(s) = 1 | + (−1.42 + 1.72i)5-s + (−3.11 + 3.11i)7-s − 1.17·11-s + (−2.15 + 2.15i)13-s + (1.33 + 1.33i)17-s + 0.322·19-s + (4.71 − 4.71i)23-s + (−0.933 − 4.91i)25-s − 6.63i·29-s + 0.0675·31-s + (−0.924 − 9.82i)35-s + (−7.60 − 7.60i)37-s − 3.19i·41-s + (−6.70 + 6.70i)43-s + (7.34 + 7.34i)47-s + ⋯ |
L(s) = 1 | + (−0.637 + 0.770i)5-s + (−1.17 + 1.17i)7-s − 0.355·11-s + (−0.596 + 0.596i)13-s + (0.323 + 0.323i)17-s + 0.0739·19-s + (0.982 − 0.982i)23-s + (−0.186 − 0.982i)25-s − 1.23i·29-s + 0.0121·31-s + (−0.156 − 1.66i)35-s + (−1.25 − 1.25i)37-s − 0.499i·41-s + (−1.02 + 1.02i)43-s + (1.07 + 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03351133512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03351133512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.42 - 1.72i)T \) |
good | 7 | \( 1 + (3.11 - 3.11i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + (2.15 - 2.15i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.33 - 1.33i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.322T + 19T^{2} \) |
| 23 | \( 1 + (-4.71 + 4.71i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 - 0.0675T + 31T^{2} \) |
| 37 | \( 1 + (7.60 + 7.60i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 + (6.70 - 6.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.73 + 5.73i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.68iT - 59T^{2} \) |
| 61 | \( 1 - 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.18iT - 71T^{2} \) |
| 73 | \( 1 + (-3.97 - 3.97i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.66iT - 79T^{2} \) |
| 83 | \( 1 + (0.585 + 0.585i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.557T + 89T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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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
−9.291683008902134484098942548046, −8.522533807125648952698670771734, −7.58485584232525329270149991332, −6.73834773240118500914830027634, −6.15212057154824883632522932706, −5.14373960283042806920844534580, −3.97645846747272166697616185320, −2.98885445751593124910792540232, −2.33147226180699236931122597242, −0.01462141945389912985469192836,
1.17886982765473592147198342943, 3.11101085415838639798800458681, 3.64533506621670294320469604327, 4.82643875791271423717716342504, 5.46123000628701447896692118211, 6.87891498779655710922241106408, 7.25319314654531477817098839727, 8.136618121226817568676985222509, 9.048165916004287401518260858563, 9.831452585321228362897184426712