| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 1.41·13-s + 16-s − 4·17-s + 2.58·19-s − 22-s + 3.65·23-s − 5·25-s − 1.41·26-s − 5.65·29-s + 11.0·31-s + 32-s − 4·34-s − 7.65·37-s + 2.58·38-s + 1.65·41-s − 10·43-s − 44-s + 3.65·46-s − 1.41·47-s − 5·50-s − 1.41·52-s − 7.65·53-s − 5.65·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 0.301·11-s − 0.392·13-s + 0.250·16-s − 0.970·17-s + 0.593·19-s − 0.213·22-s + 0.762·23-s − 25-s − 0.277·26-s − 1.05·29-s + 1.98·31-s + 0.176·32-s − 0.685·34-s − 1.25·37-s + 0.419·38-s + 0.258·41-s − 1.52·43-s − 0.150·44-s + 0.539·46-s − 0.206·47-s − 0.707·50-s − 0.196·52-s − 1.05·53-s − 0.742·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 - 0.928T + 83T^{2} \) |
| 89 | \( 1 + 5.41T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.09913476350978606604928664483, −6.71609151954282017318310140055, −5.88928797115136615471746419813, −5.17181292767253348761792101856, −4.66283279238377383597084199321, −3.84091260915414554593211408866, −3.07157399126680664505266356566, −2.33211737022148911476889012878, −1.43068313812927773263926636795, 0,
1.43068313812927773263926636795, 2.33211737022148911476889012878, 3.07157399126680664505266356566, 3.84091260915414554593211408866, 4.66283279238377383597084199321, 5.17181292767253348761792101856, 5.88928797115136615471746419813, 6.71609151954282017318310140055, 7.09913476350978606604928664483
