| L(s) = 1 | + 2.08·2-s + 3-s + 2.35·4-s + 0.146·5-s + 2.08·6-s − 2.80·7-s + 0.737·8-s + 9-s + 0.305·10-s − 2.41·11-s + 2.35·12-s + 13-s − 5.84·14-s + 0.146·15-s − 3.16·16-s − 7.90·17-s + 2.08·18-s − 1.37·19-s + 0.344·20-s − 2.80·21-s − 5.04·22-s + 2.00·23-s + 0.737·24-s − 4.97·25-s + 2.08·26-s + 27-s − 6.59·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.17·4-s + 0.0653·5-s + 0.851·6-s − 1.05·7-s + 0.260·8-s + 0.333·9-s + 0.0964·10-s − 0.729·11-s + 0.679·12-s + 0.277·13-s − 1.56·14-s + 0.0377·15-s − 0.792·16-s − 1.91·17-s + 0.491·18-s − 0.314·19-s + 0.0769·20-s − 0.611·21-s − 1.07·22-s + 0.418·23-s + 0.150·24-s − 0.995·25-s + 0.409·26-s + 0.192·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.146T + 5T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 17 | \( 1 + 7.90T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 0.323T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 8.53T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 + 3.63T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 + 5.22T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.84T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−7.947203335114815043833809883323, −7.07500538619139074600190793541, −6.37583520543221756424995403464, −5.91497291621572037943580009500, −4.85759349293074503596881566978, −4.25336939995428962035864962354, −3.47585419942920037644568779881, −2.74243520423486951971491910513, −2.06462271168688633893455251201, 0,
2.06462271168688633893455251201, 2.74243520423486951971491910513, 3.47585419942920037644568779881, 4.25336939995428962035864962354, 4.85759349293074503596881566978, 5.91497291621572037943580009500, 6.37583520543221756424995403464, 7.07500538619139074600190793541, 7.947203335114815043833809883323
