| L(s) = 1 | − 1.51·2-s − 3-s + 0.300·4-s − 2.76·5-s + 1.51·6-s − 7-s + 2.57·8-s + 9-s + 4.18·10-s − 4.10·11-s − 0.300·12-s − 1.11·13-s + 1.51·14-s + 2.76·15-s − 4.51·16-s + 4.01·17-s − 1.51·18-s − 3.77·19-s − 0.828·20-s + 21-s + 6.21·22-s + 6.00·23-s − 2.57·24-s + 2.62·25-s + 1.69·26-s − 27-s − 0.300·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.150·4-s − 1.23·5-s + 0.619·6-s − 0.377·7-s + 0.911·8-s + 0.333·9-s + 1.32·10-s − 1.23·11-s − 0.0866·12-s − 0.310·13-s + 0.405·14-s + 0.713·15-s − 1.12·16-s + 0.973·17-s − 0.357·18-s − 0.865·19-s − 0.185·20-s + 0.218·21-s + 1.32·22-s + 1.25·23-s − 0.526·24-s + 0.525·25-s + 0.332·26-s − 0.192·27-s − 0.0567·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 + 0.253T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 - 0.309T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 + 0.569T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 - 8.17T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 0.647T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.53474496932817765186002627949, −7.20940896108753473323832939390, −6.27367430996259797827360419060, −5.23111436139303682917106478921, −4.80538987639256809677653817193, −3.88739297085199377080522082939, −3.13839529486557636592725392774, −1.96297308896674279287145481240, −0.72080313612399860598837661254, 0,
0.72080313612399860598837661254, 1.96297308896674279287145481240, 3.13839529486557636592725392774, 3.88739297085199377080522082939, 4.80538987639256809677653817193, 5.23111436139303682917106478921, 6.27367430996259797827360419060, 7.20940896108753473323832939390, 7.53474496932817765186002627949
