| L(s) = 1 | − 1.60·2-s + 3-s + 0.590·4-s − 1.60·6-s − 2.89·7-s + 2.26·8-s + 9-s − 2.05·11-s + 0.590·12-s − 4.52·13-s + 4.65·14-s − 4.83·16-s + 4.14·17-s − 1.60·18-s + 3.21·19-s − 2.89·21-s + 3.30·22-s − 2.15·23-s + 2.26·24-s + 7.28·26-s + 27-s − 1.70·28-s + 9.10·29-s + 0.697·31-s + 3.24·32-s − 2.05·33-s − 6.67·34-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.577·3-s + 0.295·4-s − 0.657·6-s − 1.09·7-s + 0.801·8-s + 0.333·9-s − 0.618·11-s + 0.170·12-s − 1.25·13-s + 1.24·14-s − 1.20·16-s + 1.00·17-s − 0.379·18-s + 0.738·19-s − 0.630·21-s + 0.703·22-s − 0.449·23-s + 0.462·24-s + 1.42·26-s + 0.192·27-s − 0.322·28-s + 1.69·29-s + 0.125·31-s + 0.573·32-s − 0.357·33-s − 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 - 0.697T + 31T^{2} \) |
| 37 | \( 1 - 8.61T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 + 8.56T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 8.88T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 0.136T + 97T^{2} \) |
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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
−8.073074058073846066038322749067, −7.77204685246078191114352881349, −7.05503021309577440775876798774, −6.19030849098906721932332720021, −5.10451740248555701510411930941, −4.32397217096789637347805097833, −3.12020347096287027280422928091, −2.55138374380944435493321749900, −1.20220945587164917344787781257, 0,
1.20220945587164917344787781257, 2.55138374380944435493321749900, 3.12020347096287027280422928091, 4.32397217096789637347805097833, 5.10451740248555701510411930941, 6.19030849098906721932332720021, 7.05503021309577440775876798774, 7.77204685246078191114352881349, 8.073074058073846066038322749067
