| L(s) = 1 | + 0.499·2-s − 0.849·3-s − 1.75·4-s + 1.04·5-s − 0.424·6-s − 1.87·8-s − 2.27·9-s + 0.521·10-s + 3.96·11-s + 1.48·12-s − 0.885·15-s + 2.56·16-s + 0.142·17-s − 1.13·18-s − 5.50·19-s − 1.82·20-s + 1.98·22-s + 4.39·23-s + 1.59·24-s − 3.91·25-s + 4.48·27-s − 8.39·29-s − 0.442·30-s + 2.84·31-s + 5.03·32-s − 3.37·33-s + 0.0710·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.490·3-s − 0.875·4-s + 0.466·5-s − 0.173·6-s − 0.662·8-s − 0.759·9-s + 0.164·10-s + 1.19·11-s + 0.429·12-s − 0.228·15-s + 0.640·16-s + 0.0344·17-s − 0.268·18-s − 1.26·19-s − 0.407·20-s + 0.422·22-s + 0.915·23-s + 0.325·24-s − 0.782·25-s + 0.863·27-s − 1.55·29-s − 0.0808·30-s + 0.511·31-s + 0.889·32-s − 0.586·33-s + 0.0121·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.499T + 2T^{2} \) |
| 3 | \( 1 + 0.849T + 3T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 17 | \( 1 - 0.142T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 8.39T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 - 0.843T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + 0.279T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 5.86T + 61T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 2.87T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.43440516219041111471255455071, −6.40378254358065809999202240005, −6.03018931942607519799469383899, −5.47411410090332889139609588659, −4.60312050517118054179199343902, −4.06151787984168965530261984481, −3.25174021278242655637725880210, −2.25976231629670613348156581994, −1.11015344611705021705922771788, 0,
1.11015344611705021705922771788, 2.25976231629670613348156581994, 3.25174021278242655637725880210, 4.06151787984168965530261984481, 4.60312050517118054179199343902, 5.47411410090332889139609588659, 6.03018931942607519799469383899, 6.40378254358065809999202240005, 7.43440516219041111471255455071
