L(s) = 1 | + (−2.23 − 0.114i)5-s − 3.07·7-s − 1.12·11-s + 3.92i·13-s − 0.286i·17-s + 5.12i·19-s + (4.17 − 2.35i)23-s + (4.97 + 0.510i)25-s + 3.70i·29-s − 5.14·31-s + (6.86 + 0.351i)35-s − 1.46·37-s + 4.70i·41-s − 4.70·43-s + 3.27·47-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0511i)5-s − 1.16·7-s − 0.338·11-s + 1.08i·13-s − 0.0693i·17-s + 1.17i·19-s + (0.870 − 0.491i)23-s + (0.994 + 0.102i)25-s + 0.688i·29-s − 0.923·31-s + (1.16 + 0.0593i)35-s − 0.240·37-s + 0.734i·41-s − 0.716·43-s + 0.477·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3661469498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3661469498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.114i)T \) |
| 23 | \( 1 + (-4.17 + 2.35i)T \) |
good | 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 - 3.92iT - 13T^{2} \) |
| 17 | \( 1 + 0.286iT - 17T^{2} \) |
| 19 | \( 1 - 5.12iT - 19T^{2} \) |
| 29 | \( 1 - 3.70iT - 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 - 4.70iT - 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 - 3.68iT - 53T^{2} \) |
| 59 | \( 1 + 1.30iT - 59T^{2} \) |
| 61 | \( 1 + 5.75iT - 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 + 1.29iT - 71T^{2} \) |
| 73 | \( 1 + 9.83iT - 73T^{2} \) |
| 79 | \( 1 - 0.954iT - 79T^{2} \) |
| 83 | \( 1 + 7.87iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.203840616967759195740901589810, −7.41950458528506733777806330125, −6.80843265212339786933490076309, −6.18048864443456736514055877838, −5.15297739943580876044422430132, −4.33007937242027266312807917216, −3.53961611136765508315473924168, −2.94720598211230045241854985056, −1.61264783884550894135229776300, −0.14608376432261966515961555118,
0.791851264673412151645620756712, 2.54750957215052232281481666967, 3.23633748427432667342523504180, 3.86799494845210943485788988551, 4.92323514633626551287825670871, 5.60303992075642931160879437654, 6.58141518020766262351854235764, 7.20745729642611087322263949853, 7.76944808603604705519612501257, 8.657337479015890011369954242444