| L(s) = 1 | − 1.19i·5-s − i·7-s − 1.62·11-s − 2.09·13-s + 0.585i·17-s − 5.89i·19-s + 7.09·23-s + 3.56·25-s − 7.08i·29-s + 6.72i·31-s − 1.19·35-s + 2.17·37-s − 0.164i·41-s + 5.75i·43-s − 3.41·47-s + ⋯ |
| L(s) = 1 | − 0.536i·5-s − 0.377i·7-s − 0.491·11-s − 0.581·13-s + 0.142i·17-s − 1.35i·19-s + 1.47·23-s + 0.712·25-s − 1.31i·29-s + 1.20i·31-s − 0.202·35-s + 0.357·37-s − 0.0256i·41-s + 0.877i·43-s − 0.498·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.225462840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225462840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.19iT - 5T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 - 0.585iT - 17T^{2} \) |
| 19 | \( 1 + 5.89iT - 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 + 7.08iT - 29T^{2} \) |
| 31 | \( 1 - 6.72iT - 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 0.164iT - 41T^{2} \) |
| 43 | \( 1 - 5.75iT - 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 3.29iT - 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 0.702iT - 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 6.33iT - 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−7.81826203052803057217035325384, −7.05340232445177798891165616744, −6.57956501100216842348765719143, −5.48391286862503603938263193861, −4.84595011740279269320612800731, −4.40764027588677753656834052436, −3.17651300696022795769662876076, −2.57153215485631563335810402541, −1.32092748651853921460799368334, −0.33316073626517574743890114605,
1.21302322791734309803042531012, 2.38564921050900223056319735961, 2.99719430008900515706932125422, 3.85431046611734685075030215825, 4.86768802581779777901662675470, 5.45974873474781357088098643323, 6.19548337867236469750684198542, 7.10432759545367246805751356143, 7.43598922637418240997050510535, 8.400413252965113134367708307334
