L(s) = 1 | + (−1.67 − 0.448i)2-s + (1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (1.67 − 0.448i)18-s + (−1.22 + 1.22i)22-s + (0.448 − 1.67i)23-s − i·29-s − 2·36-s + (1.67 + 0.448i)37-s + (1.22 + 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
L(s) = 1 | + (−1.67 − 0.448i)2-s + (1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (1.67 − 0.448i)18-s + (−1.22 + 1.22i)22-s + (0.448 − 1.67i)23-s − i·29-s − 2·36-s + (1.67 + 0.448i)37-s + (1.22 + 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4493610666\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4493610666\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.602043827071134545731981088649, −9.045220977357553702149470134919, −8.207813644904337253833370774792, −7.87476645177958270627845876133, −6.63312760434713522313148700277, −5.95275891675322451766954676615, −4.53426284451375565156702819608, −3.06351547788081039922688455579, −2.31278192804378191283822101731, −0.795073279066333925922302132925,
1.19578509012443993254680863876, 2.46475113049638324653093542350, 3.84273784607651024876426960417, 5.35309891183750941152610222615, 6.20158632784032086414084765505, 7.08478272532707150749643069449, 7.63137124690433274908751214152, 8.591689426133857874931771599756, 9.278278982576833044602296419632, 9.642920884418739934945132279550