L(s) = 1 | + (0.923 + 0.923i)3-s + (2.48 − 0.915i)7-s − 1.29i·9-s + 1.21·11-s + (0.996 + 0.996i)13-s + (0.567 − 0.567i)17-s − 0.103·19-s + (3.13 + 1.44i)21-s + (1.43 − 1.43i)23-s + (3.96 − 3.96i)27-s − 4.97i·29-s + 4.35i·31-s + (1.12 + 1.12i)33-s + (−2.54 − 2.54i)37-s + 1.83i·39-s + ⋯ |
L(s) = 1 | + (0.532 + 0.532i)3-s + (0.938 − 0.345i)7-s − 0.431i·9-s + 0.366·11-s + (0.276 + 0.276i)13-s + (0.137 − 0.137i)17-s − 0.0237·19-s + (0.684 + 0.315i)21-s + (0.298 − 0.298i)23-s + (0.763 − 0.763i)27-s − 0.924i·29-s + 0.781i·31-s + (0.195 + 0.195i)33-s + (−0.417 − 0.417i)37-s + 0.294i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.354325465\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.354325465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.48 + 0.915i)T \) |
good | 3 | \( 1 + (-0.923 - 0.923i)T + 3iT^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + (-0.996 - 0.996i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.567 + 0.567i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.103T + 19T^{2} \) |
| 23 | \( 1 + (-1.43 + 1.43i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 - 4.35iT - 31T^{2} \) |
| 37 | \( 1 + (2.54 + 2.54i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.06iT - 41T^{2} \) |
| 43 | \( 1 + (-3.11 + 3.11i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.23 + 7.23i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.882 + 0.882i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 + (7.07 + 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.329T + 71T^{2} \) |
| 73 | \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \) |
| 79 | \( 1 - 13.6iT - 79T^{2} \) |
| 83 | \( 1 + (-4.61 - 4.61i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + (2.40 - 2.40i)T - 97iT^{2} \) |
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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
−9.455938431213634195246369203793, −8.799064613122925419333800524362, −8.134924395622983323881129263535, −7.19418821496050971574837642568, −6.34208374165087736116379492669, −5.23718903118891781333463049878, −4.29896649625542274169146110471, −3.67024803078235074771773146215, −2.45726480125891447067033410005, −1.09679308713651853307698736737,
1.32389342409451089672067900574, 2.23540048129232389239861714598, 3.32420226319807841272320617386, 4.54206007811450403078444896184, 5.36553135318048531238748369554, 6.30801532729708180044084901160, 7.45294843183479727215862252735, 7.83670827042299396700105752287, 8.738529168290621622216152857533, 9.248906296538156127146648527852