| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s − 2.61·5-s − 1.61·6-s − 7-s + 2.23·8-s + 9-s + 4.23·10-s + 0.618·12-s + 13-s + 1.61·14-s − 2.61·15-s − 4.85·16-s − 0.236·17-s − 1.61·18-s − 1.61·20-s − 21-s + 1.23·23-s + 2.23·24-s + 1.85·25-s − 1.61·26-s + 27-s − 0.618·28-s + 6.70·29-s + 4.23·30-s − 5.76·31-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s − 1.17·5-s − 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 1.33·10-s + 0.178·12-s + 0.277·13-s + 0.432·14-s − 0.675·15-s − 1.21·16-s − 0.0572·17-s − 0.381·18-s − 0.361·20-s − 0.218·21-s + 0.257·23-s + 0.456·24-s + 0.370·25-s − 0.317·26-s + 0.192·27-s − 0.116·28-s + 1.24·29-s + 0.773·30-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 8.56T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−8.589058783836028786737727170165, −7.929801309632304516973770464508, −7.30159053092529634177296414482, −6.68737466062540835395838319970, −5.32936807357871981073622536746, −4.25106368741350336165207209461, −3.67786734473746478214096498023, −2.55164948989479563206290520385, −1.22920402494082472663285759751, 0,
1.22920402494082472663285759751, 2.55164948989479563206290520385, 3.67786734473746478214096498023, 4.25106368741350336165207209461, 5.32936807357871981073622536746, 6.68737466062540835395838319970, 7.30159053092529634177296414482, 7.929801309632304516973770464508, 8.589058783836028786737727170165
