| L(s) = 1 | − 0.556·2-s + 2.53·3-s − 1.69·4-s + 0.874·5-s − 1.41·6-s + 4.50·7-s + 2.05·8-s + 3.44·9-s − 0.486·10-s + 11-s − 4.29·12-s + 1.21·13-s − 2.50·14-s + 2.22·15-s + 2.23·16-s − 6.11·17-s − 1.91·18-s − 3.45·19-s − 1.47·20-s + 11.4·21-s − 0.556·22-s − 1.78·23-s + 5.21·24-s − 4.23·25-s − 0.674·26-s + 1.12·27-s − 7.61·28-s + ⋯ |
| L(s) = 1 | − 0.393·2-s + 1.46·3-s − 0.845·4-s + 0.391·5-s − 0.576·6-s + 1.70·7-s + 0.726·8-s + 1.14·9-s − 0.153·10-s + 0.301·11-s − 1.23·12-s + 0.336·13-s − 0.669·14-s + 0.573·15-s + 0.559·16-s − 1.48·17-s − 0.451·18-s − 0.791·19-s − 0.330·20-s + 2.49·21-s − 0.118·22-s − 0.371·23-s + 1.06·24-s − 0.846·25-s − 0.132·26-s + 0.217·27-s − 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075216213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075216213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.556T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 0.874T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−10.34513445567515072154341008877, −9.221112285070898554663779360738, −8.810781576095355041214864163343, −8.133390281098717015996635457772, −7.55418817571703716381678493692, −6.03349074054235155343894651403, −4.52674260335720068990824510734, −4.20427234627109125637011425201, −2.48767590512141364555885786388, −1.51155511663515700733740461446,
1.51155511663515700733740461446, 2.48767590512141364555885786388, 4.20427234627109125637011425201, 4.52674260335720068990824510734, 6.03349074054235155343894651403, 7.55418817571703716381678493692, 8.133390281098717015996635457772, 8.810781576095355041214864163343, 9.221112285070898554663779360738, 10.34513445567515072154341008877
