| L(s) = 1 | − 3.16i·3-s − 2.23i·5-s + (2.12 − 1.58i)7-s − 7.00·9-s − 7.07·15-s + (−5.00 − 6.70i)21-s + 1.41·23-s − 5.00·25-s + 12.6i·27-s − 6·29-s + (−3.53 − 4.74i)35-s − 4.47i·41-s − 12.7·43-s + 15.6i·45-s + 9.48i·47-s + ⋯ |
| L(s) = 1 | − 1.82i·3-s − 0.999i·5-s + (0.801 − 0.597i)7-s − 2.33·9-s − 1.82·15-s + (−1.09 − 1.46i)21-s + 0.294·23-s − 1.00·25-s + 2.43i·27-s − 1.11·29-s + (−0.597 − 0.801i)35-s − 0.698i·41-s − 1.94·43-s + 2.33i·45-s + 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.248227887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248227887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (-2.12 + 1.58i)T \) |
| good | 3 | \( 1 + 3.16iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 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
−8.342028285786165593201658036370, −7.72575287485226003618314501301, −7.18028082369352147729478987760, −6.29553787114324606773941569934, −5.45685524677552173695102718890, −4.69895265053037693066644683287, −3.45524386553575890633797234539, −2.05046882990868582126221356881, −1.43801247716543431557119253970, −0.42169270087196027488623536318,
2.13066588276179213650084271881, 3.13025275255520846070016035611, 3.81536893447775702771458449438, 4.75633790830764163800903661032, 5.40967421688039287204219122538, 6.15960520351229691892557538214, 7.26973583109401719430566055248, 8.265284532330633544362084793775, 8.862237595193860223723517390904, 9.684222130624878434782256366141
