| L(s) = 1 | + 3.41·3-s + 8.65·9-s − 2.24·11-s − 5.65·17-s + 4.58·19-s − 5·25-s + 19.3·27-s − 7.65·33-s − 6·41-s − 1.07·43-s − 7·49-s − 19.3·51-s + 15.6·57-s − 5.75·59-s + 15.8·67-s − 16.9·73-s − 17.0·75-s + 39.9·81-s + 14.7·83-s + 5.65·89-s + 16.9·97-s − 19.4·99-s − 18.2·107-s − 18·113-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 2.88·9-s − 0.676·11-s − 1.37·17-s + 1.05·19-s − 25-s + 3.71·27-s − 1.33·33-s − 0.937·41-s − 0.163·43-s − 49-s − 2.70·51-s + 2.07·57-s − 0.749·59-s + 1.94·67-s − 1.98·73-s − 1.97·75-s + 4.44·81-s + 1.61·83-s + 0.599·89-s + 1.72·97-s − 1.95·99-s − 1.76·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.640292317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.640292317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−10.60417879617760738963477666130, −9.735398418577798032356123542837, −9.090498711873451502138258335141, −8.210915151021506824382224722761, −7.59261692385563935422250155263, −6.64112494412112710620413980932, −4.94269025013754950277331619420, −3.85552847733976615835611606496, −2.87309189232451863575736962880, −1.86215702794977185136290616837,
1.86215702794977185136290616837, 2.87309189232451863575736962880, 3.85552847733976615835611606496, 4.94269025013754950277331619420, 6.64112494412112710620413980932, 7.59261692385563935422250155263, 8.210915151021506824382224722761, 9.090498711873451502138258335141, 9.735398418577798032356123542837, 10.60417879617760738963477666130
