| L(s) = 1 | + 2.50·2-s + 1.03·3-s + 4.28·4-s − 5-s + 2.59·6-s + 5.73·8-s − 1.92·9-s − 2.50·10-s − 3.75·11-s + 4.43·12-s − 1.79·13-s − 1.03·15-s + 5.81·16-s − 0.478·17-s − 4.83·18-s − 5.10·19-s − 4.28·20-s − 9.42·22-s + 5.65·23-s + 5.94·24-s + 25-s − 4.49·26-s − 5.10·27-s + 2.86·29-s − 2.59·30-s − 3.89·31-s + 3.10·32-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.597·3-s + 2.14·4-s − 0.447·5-s + 1.05·6-s + 2.02·8-s − 0.642·9-s − 0.793·10-s − 1.13·11-s + 1.28·12-s − 0.497·13-s − 0.267·15-s + 1.45·16-s − 0.116·17-s − 1.14·18-s − 1.17·19-s − 0.959·20-s − 2.00·22-s + 1.17·23-s + 1.21·24-s + 0.200·25-s − 0.881·26-s − 0.981·27-s + 0.531·29-s − 0.473·30-s − 0.700·31-s + 0.549·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 1.03T + 3T^{2} \) |
| 11 | \( 1 + 3.75T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 0.478T + 17T^{2} \) |
| 19 | \( 1 + 5.10T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 1.62T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.02623047793552053070038257371, −6.76897857770060229521263093654, −5.69703655918182281266123097909, −5.23553731401941890141810278713, −4.60253442226059266664982300975, −3.84939310892354261665027567595, −3.06863177767160306256372610889, −2.66676727104721738711218679477, −1.86180844359285234271755741735, 0,
1.86180844359285234271755741735, 2.66676727104721738711218679477, 3.06863177767160306256372610889, 3.84939310892354261665027567595, 4.60253442226059266664982300975, 5.23553731401941890141810278713, 5.69703655918182281266123097909, 6.76897857770060229521263093654, 7.02623047793552053070038257371
