L(s) = 1 | + (0.309 − 0.951i)5-s + (0.809 − 0.587i)9-s + (1.11 + 1.53i)13-s + (0.5 − 1.53i)17-s + (−0.809 − 0.587i)25-s + (−1.11 + 0.363i)29-s + (0.690 + 0.951i)37-s + (−0.5 + 0.363i)41-s + (−0.309 − 0.951i)45-s + 49-s + (−1.80 + 0.587i)53-s + (1.11 − 1.53i)61-s + (1.80 − 0.587i)65-s + (0.5 + 0.363i)73-s + (0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)5-s + (0.809 − 0.587i)9-s + (1.11 + 1.53i)13-s + (0.5 − 1.53i)17-s + (−0.809 − 0.587i)25-s + (−1.11 + 0.363i)29-s + (0.690 + 0.951i)37-s + (−0.5 + 0.363i)41-s + (−0.309 − 0.951i)45-s + 49-s + (−1.80 + 0.587i)53-s + (1.11 − 1.53i)61-s + (1.80 − 0.587i)65-s + (0.5 + 0.363i)73-s + (0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.494030746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494030746\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−8.899958643552260211637312945887, −8.115995678794979033451498137407, −7.16658739038973400093649343759, −6.55217397703388180403688745843, −5.72723660637685754526768574144, −4.80069433711175601781634383879, −4.20737774052482521239816982968, −3.30345213743711128402589993621, −1.88795863499926461856538579328, −1.08394966829981471656230195669,
1.40277835620315055846123417587, 2.39610336335909866678926571752, 3.50883291287921455210972839356, 3.99453261065759528227823222209, 5.38780688575099184344013321589, 5.88437563566749590847872123990, 6.63068023174151641243753011140, 7.64727478356739997118431796568, 7.932121859423991181769263572763, 8.888623130732186552172015450641