| L(s) = 1 | − 2.10·2-s + 1.73·3-s + 2.44·4-s + 0.240·5-s − 3.65·6-s + 0.0249·7-s − 0.934·8-s + 0.00822·9-s − 0.507·10-s − 11-s + 4.23·12-s − 5.36·13-s − 0.0525·14-s + 0.417·15-s − 2.91·16-s + 5.56·17-s − 0.0173·18-s + 3.35·19-s + 0.587·20-s + 0.0431·21-s + 2.10·22-s − 7.08·23-s − 1.62·24-s − 4.94·25-s + 11.3·26-s − 5.18·27-s + 0.0608·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.00·3-s + 1.22·4-s + 0.107·5-s − 1.49·6-s + 0.00941·7-s − 0.330·8-s + 0.00274·9-s − 0.160·10-s − 0.301·11-s + 1.22·12-s − 1.48·13-s − 0.0140·14-s + 0.107·15-s − 0.729·16-s + 1.35·17-s − 0.00408·18-s + 0.770·19-s + 0.131·20-s + 0.00942·21-s + 0.449·22-s − 1.47·23-s − 0.330·24-s − 0.988·25-s + 2.21·26-s − 0.998·27-s + 0.0115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 0.240T + 5T^{2} \) |
| 7 | \( 1 - 0.0249T + 7T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 73 | \( 1 - 6.84T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 + 5.73T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 - 7.83T + 97T^{2} \) |
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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
−9.370808544601287323056674654000, −8.068748865578257226283316051801, −7.955933382326157291593926311517, −7.30054155482558783253007400160, −6.03253245520275451386768199479, −4.97253831068498225377654921446, −3.59901877186884686558238086695, −2.56135976223944539541697872770, −1.70820259457270093862890357055, 0,
1.70820259457270093862890357055, 2.56135976223944539541697872770, 3.59901877186884686558238086695, 4.97253831068498225377654921446, 6.03253245520275451386768199479, 7.30054155482558783253007400160, 7.955933382326157291593926311517, 8.068748865578257226283316051801, 9.370808544601287323056674654000
