L(s) = 1 | + (−1.73 − 1.73i)5-s + 1.03i·7-s + (0.896 + 0.896i)11-s + (−3.73 + 3.73i)13-s + 3.46·17-s + (0.896 − 0.896i)19-s − 6.69i·23-s + 0.999i·25-s + (−1.73 + 1.73i)29-s + 5.65·31-s + (1.79 − 1.79i)35-s + (0.267 + 0.267i)37-s − 6.92i·41-s + (−5.79 − 5.79i)43-s − 9.79·47-s + ⋯ |
L(s) = 1 | + (−0.774 − 0.774i)5-s + 0.391i·7-s + (0.270 + 0.270i)11-s + (−1.03 + 1.03i)13-s + 0.840·17-s + (0.205 − 0.205i)19-s − 1.39i·23-s + 0.199i·25-s + (−0.321 + 0.321i)29-s + 1.01·31-s + (0.303 − 0.303i)35-s + (0.0440 + 0.0440i)37-s − 1.08i·41-s + (−0.883 − 0.883i)43-s − 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7312265585\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7312265585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.73 + 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.03iT - 7T^{2} \) |
| 11 | \( 1 + (-0.896 - 0.896i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.73 - 3.73i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + (-0.896 + 0.896i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.69iT - 23T^{2} \) |
| 29 | \( 1 + (1.73 - 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-0.267 - 0.267i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (5.79 + 5.79i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (4.26 + 4.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.58 + 7.58i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.267 + 0.267i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.96 + 2.96i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.69iT - 71T^{2} \) |
| 73 | \( 1 - 9.46iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (5.79 - 5.79i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−8.601208420570751619498497160274, −8.149223307430476328425819531235, −7.16205786994347594674307172855, −6.56715051159678816357157024402, −5.36683326574130369597686939056, −4.68685240573040356618145280284, −4.03275173161744639496082196429, −2.84854582067871515844140343484, −1.72098155555150119195472034498, −0.26675520894466077344090085972,
1.26640962308645660785452489062, 2.92625435318348320299439099048, 3.34576915910385019348630522339, 4.39485482625261281712139172864, 5.35386214035435746797494383257, 6.19809426965172397328062432709, 7.20012886949048211399376175561, 7.68834780352888674427392090041, 8.197144210689934104148832592798, 9.480059396002720262567149142550