| L(s) = 1 | + 2-s − 0.138·3-s + 4-s − 3.80·5-s − 0.138·6-s + 3.68·7-s + 8-s − 2.98·9-s − 3.80·10-s + 4.03·11-s − 0.138·12-s − 4.85·13-s + 3.68·14-s + 0.525·15-s + 16-s − 1.82·17-s − 2.98·18-s − 1.46·19-s − 3.80·20-s − 0.509·21-s + 4.03·22-s + 6.78·23-s − 0.138·24-s + 9.46·25-s − 4.85·26-s + 0.826·27-s + 3.68·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0797·3-s + 0.5·4-s − 1.70·5-s − 0.0564·6-s + 1.39·7-s + 0.353·8-s − 0.993·9-s − 1.20·10-s + 1.21·11-s − 0.0398·12-s − 1.34·13-s + 0.985·14-s + 0.135·15-s + 0.250·16-s − 0.443·17-s − 0.702·18-s − 0.335·19-s − 0.850·20-s − 0.111·21-s + 0.860·22-s + 1.41·23-s − 0.0282·24-s + 1.89·25-s − 0.952·26-s + 0.159·27-s + 0.697·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.138T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.78T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 + 0.900T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.72T + 67T^{2} \) |
| 71 | \( 1 + 6.54T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.78688624447459288090522310842, −7.45140702375486238709683880398, −6.75256240297230719383909471854, −5.62840635501648349218252658791, −4.81825768201391645099832701756, −4.40117420438684221332963727109, −3.57761906186536061927263896957, −2.70543263976121423893249791337, −1.49813647437497609933718994391, 0,
1.49813647437497609933718994391, 2.70543263976121423893249791337, 3.57761906186536061927263896957, 4.40117420438684221332963727109, 4.81825768201391645099832701756, 5.62840635501648349218252658791, 6.75256240297230719383909471854, 7.45140702375486238709683880398, 7.78688624447459288090522310842
