L(s) = 1 | + (0.662 − 0.382i)2-s + (0.258 − 0.965i)3-s + (−0.207 + 0.358i)4-s + (0.382 + 0.923i)5-s + (−0.198 − 0.739i)6-s + 1.08i·8-s + (−0.866 − 0.499i)9-s + (0.607 + 0.465i)10-s + (−0.478 + 1.78i)11-s + (0.292 + 0.292i)12-s + (0.991 − 0.130i)15-s + (0.207 + 0.358i)16-s − 0.765·18-s + (−0.410 − 0.0540i)20-s + (0.366 + 1.36i)22-s + ⋯ |
L(s) = 1 | + (0.662 − 0.382i)2-s + (0.258 − 0.965i)3-s + (−0.207 + 0.358i)4-s + (0.382 + 0.923i)5-s + (−0.198 − 0.739i)6-s + 1.08i·8-s + (−0.866 − 0.499i)9-s + (0.607 + 0.465i)10-s + (−0.478 + 1.78i)11-s + (0.292 + 0.292i)12-s + (0.991 − 0.130i)15-s + (0.207 + 0.358i)16-s − 0.765·18-s + (−0.410 − 0.0540i)20-s + (0.366 + 1.36i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.612791311\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612791311\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - 1.84T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.84T + T^{2} \) |
| 89 | \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.209924012514817227359593093387, −8.148046626505307892532990121906, −7.53593717215765917589996942250, −6.99133980776036961503145454464, −6.10508803221941796809394192623, −5.21183732579918353749885898716, −4.28978325843952143536284275883, −3.27357578129511673128187412438, −2.46162577345754769145894701738, −1.89765510684987995639467128385,
0.832948678789467777371635779528, 2.56762357036096503206517098428, 3.68095866274124524745432394535, 4.29031175306230883078188697372, 5.25507676202726150967328085739, 5.62746523600488904585729715457, 6.22925175467704255480182469116, 7.58512641483014882777623827520, 8.548007001313629620163494901172, 8.961377089828042891364500620272