| L(s) = 1 | + 3.13·3-s − 0.594·5-s − 3.48·7-s + 6.83·9-s − 4.83·11-s − 0.215·13-s − 1.86·15-s + 1.29·17-s + 19-s − 10.9·21-s + 4.52·23-s − 4.64·25-s + 12.0·27-s − 9.41·29-s + 1.22·31-s − 15.1·33-s + 2.07·35-s − 5.62·37-s − 0.676·39-s + 0.450·41-s + 0.794·43-s − 4.06·45-s − 12.1·47-s + 5.16·49-s + 4.06·51-s − 2.56·53-s + 2.87·55-s + ⋯ |
| L(s) = 1 | + 1.81·3-s − 0.265·5-s − 1.31·7-s + 2.27·9-s − 1.45·11-s − 0.0597·13-s − 0.481·15-s + 0.314·17-s + 0.229·19-s − 2.38·21-s + 0.944·23-s − 0.929·25-s + 2.31·27-s − 1.74·29-s + 0.219·31-s − 2.63·33-s + 0.350·35-s − 0.924·37-s − 0.108·39-s + 0.0702·41-s + 0.121·43-s − 0.605·45-s − 1.77·47-s + 0.737·49-s + 0.569·51-s − 0.352·53-s + 0.387·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.594T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 + 0.215T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 5.62T + 37T^{2} \) |
| 41 | \( 1 - 0.450T + 41T^{2} \) |
| 43 | \( 1 - 0.794T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−7.85771874840515909456257200830, −7.47081590028984465476335114322, −6.76241079701766293673298558166, −5.72024583081971674201512733217, −4.79986320104781523762208227490, −3.72931181382863528202963076006, −3.24768476099019675280658918047, −2.67359820807535463040880148434, −1.72548033297162996357982665709, 0,
1.72548033297162996357982665709, 2.67359820807535463040880148434, 3.24768476099019675280658918047, 3.72931181382863528202963076006, 4.79986320104781523762208227490, 5.72024583081971674201512733217, 6.76241079701766293673298558166, 7.47081590028984465476335114322, 7.85771874840515909456257200830
