| L(s) = 1 | − 3.31·3-s + 1.89·5-s − 7-s + 7.97·9-s + 4.72·13-s − 6.29·15-s + 4.68·17-s + 6.55·19-s + 3.31·21-s − 7.66·23-s − 1.39·25-s − 16.4·27-s − 5.21·29-s − 4.68·31-s − 1.89·35-s − 1.82·37-s − 15.6·39-s − 9.27·41-s + 2.41·43-s + 15.1·45-s − 3.31·47-s + 49-s − 15.5·51-s − 11.2·53-s − 21.7·57-s − 0.0709·59-s − 3.75·61-s + ⋯ |
| L(s) = 1 | − 1.91·3-s + 0.849·5-s − 0.377·7-s + 2.65·9-s + 1.31·13-s − 1.62·15-s + 1.13·17-s + 1.50·19-s + 0.723·21-s − 1.59·23-s − 0.278·25-s − 3.17·27-s − 0.967·29-s − 0.841·31-s − 0.321·35-s − 0.300·37-s − 2.50·39-s − 1.44·41-s + 0.368·43-s + 2.25·45-s − 0.483·47-s + 0.142·49-s − 2.17·51-s − 1.54·53-s − 2.87·57-s − 0.00924·59-s − 0.481·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.72T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 7.66T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 0.0709T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 1.37T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 0.100T + 97T^{2} \) |
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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
−7.40666457729963315301589639463, −6.53798432110698029677621869540, −6.09445769735839342464429268382, −5.47531607282890197118241344482, −5.23406907850318415224510876975, −3.98693371964154446139873693453, −3.42842930284893534689982771207, −1.78332905627953225213683655995, −1.22984106661356425606915174082, 0,
1.22984106661356425606915174082, 1.78332905627953225213683655995, 3.42842930284893534689982771207, 3.98693371964154446139873693453, 5.23406907850318415224510876975, 5.47531607282890197118241344482, 6.09445769735839342464429268382, 6.53798432110698029677621869540, 7.40666457729963315301589639463
