| L(s) = 1 | − 2.66·2-s + 5.09·4-s − 0.0170·7-s − 8.24·8-s − 1.70·11-s − 4.69·13-s + 0.0453·14-s + 11.7·16-s + 3.91·17-s + 6.02·19-s + 4.55·22-s − 4.82·23-s + 12.5·26-s − 0.0868·28-s + 29-s + 1.33·31-s − 14.8·32-s − 10.4·34-s − 8.18·37-s − 16.0·38-s − 8.36·41-s + 3.72·43-s − 8.71·44-s + 12.8·46-s + 5.36·47-s − 6.99·49-s − 23.9·52-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.54·4-s − 0.00644·7-s − 2.91·8-s − 0.515·11-s − 1.30·13-s + 0.0121·14-s + 2.94·16-s + 0.949·17-s + 1.38·19-s + 0.971·22-s − 1.00·23-s + 2.45·26-s − 0.0164·28-s + 0.185·29-s + 0.240·31-s − 2.62·32-s − 1.78·34-s − 1.34·37-s − 2.60·38-s − 1.30·41-s + 0.568·43-s − 1.31·44-s + 1.89·46-s + 0.783·47-s − 0.999·49-s − 3.31·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 + 0.0170T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 + 8.18T + 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 - 5.72T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 - 0.233T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.915806525853317785872822875458, −7.16730154904793843924436211815, −6.73208390106931941567976193168, −5.61722730039335985952213258582, −5.13645935098172648585929567846, −3.65092038505664532493467402809, −2.77742429154066799543385515355, −2.06617932622121829615956977042, −1.05002558935348783221617270892, 0,
1.05002558935348783221617270892, 2.06617932622121829615956977042, 2.77742429154066799543385515355, 3.65092038505664532493467402809, 5.13645935098172648585929567846, 5.61722730039335985952213258582, 6.73208390106931941567976193168, 7.16730154904793843924436211815, 7.915806525853317785872822875458
