| L(s) = 1 | − 2.82·5-s + (−2.44 + i)7-s + 3.46·11-s + 3.00·25-s + 10.3i·29-s − 4.89·31-s + (6.92 − 2.82i)35-s + (4.99 − 4.89i)49-s − 3.46i·53-s − 9.79·55-s + 11.3i·59-s − 9.79i·73-s + (−8.48 + 3.46i)77-s − 10i·79-s + 5.65i·83-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + (−0.925 + 0.377i)7-s + 1.04·11-s + 0.600·25-s + 1.92i·29-s − 0.879·31-s + (1.17 − 0.478i)35-s + (0.714 − 0.699i)49-s − 0.475i·53-s − 1.32·55-s + 1.47i·59-s − 1.14i·73-s + (−0.966 + 0.394i)77-s − 1.12i·79-s + 0.620i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4068071230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4068071230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.44 - i)T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19.5iT - 97T^{2} \) |
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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
−8.282658478739472797908220100070, −7.32688190772062539634319667653, −6.90835290261973504808501668550, −6.09479640960748416544866308205, −5.20689545487509055929390850793, −4.20203160594071829036102271990, −3.59498307625379655148398110851, −2.93761556411439944380729780116, −1.52801392550363271480735967193, −0.14993559268663403709448646835,
0.928397280655185399272894274449, 2.40261347808898621277039811410, 3.62278595367242572538973948387, 3.83949679245308112474813870662, 4.69821182975906450204293328393, 5.89059044934991028206232659015, 6.55246512372151691215874846562, 7.25027400032813191822835743990, 7.86306767483438054799365683651, 8.606655831469089886197960006891
