| L(s) = 1 | + 4.11·5-s − 1.10·11-s + 3.60i·13-s + 11.9·25-s + 7.82i·41-s + 12.4i·43-s − 10.0i·47-s + 7·49-s − 4.53·55-s + 15.3·59-s − 7.21i·61-s + 14.8i·65-s + 16.1i·71-s + 10.3·79-s + 13.1·83-s + ⋯ |
| L(s) = 1 | + 1.84·5-s − 0.332·11-s + 0.999i·13-s + 2.38·25-s + 1.22i·41-s + 1.90i·43-s − 1.46i·47-s + 49-s − 0.611·55-s + 1.99·59-s − 0.923i·61-s + 1.84i·65-s + 1.91i·71-s + 1.16·79-s + 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.734289211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734289211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 3.60iT \) |
| good | 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 7.82iT - 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 15.3T + 59T^{2} \) |
| 61 | \( 1 + 7.21iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 4.31iT - 89T^{2} \) |
| 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
−8.743192565150064928307536740916, −7.914488623599160292864372818146, −6.73973310478474019800549678282, −6.49598900621044597434191629717, −5.53487645596419794712574544892, −5.03893801406570809448987933012, −4.02218037823387671302952218371, −2.77121206467707983942215111245, −2.11011143810183688354595020779, −1.20229384528900581483701988726,
0.855153799797774257864266457207, 2.04520570505250341512213668972, 2.64487096164307632119016345909, 3.70202216907779536168867621392, 4.98130923164868668010409336309, 5.51028147830871806509145829335, 6.05423559753166710480350597549, 6.88169498530615894612983191419, 7.66882898249937098850254977816, 8.645838821877081560549968251353
