L(s) = 1 | + (−1 + 1.41i)3-s + (−1.00 − 2.82i)9-s + 4.89·11-s − 4.89·13-s + 3.46i·17-s + 3.46i·19-s + 6·23-s + (5.00 + 1.41i)27-s + 2.82i·29-s + 3.46i·31-s + (−4.89 + 6.92i)33-s − 4.89·37-s + (4.89 − 6.92i)39-s − 5.65i·41-s + 8.48i·43-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (−0.333 − 0.942i)9-s + 1.47·11-s − 1.35·13-s + 0.840i·17-s + 0.794i·19-s + 1.25·23-s + (0.962 + 0.272i)27-s + 0.525i·29-s + 0.622i·31-s + (−0.852 + 1.20i)33-s − 0.805·37-s + (0.784 − 1.10i)39-s − 0.883i·41-s + 1.29i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.085411040\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085411040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.07503704680591695131954225301, −9.215472541076690350721456504951, −8.716224184455137075058238657893, −7.37759934495522948997820154767, −6.60699832504716808653534874994, −5.75036462864577452775225037478, −4.82042653783507017369988367564, −4.03814327840794976747259224922, −3.07623460161938996424843183634, −1.37448176811266162374016199869,
0.53832429936472359896102591639, 1.90662655269587847477109080815, 3.04213165138257558841128148061, 4.53496931017069318699897472810, 5.21093337067728124045211522524, 6.33167253430315769380062780932, 7.02837343258190653507048087634, 7.52145958466092279660866810092, 8.750828281244510916600226839452, 9.406124878510648576761274023550