| L(s) = 1 | − 2-s − 2.79·3-s + 4-s − 5-s + 2.79·6-s + 4.83·7-s − 8-s + 4.79·9-s + 10-s − 4.34·11-s − 2.79·12-s − 4.83·14-s + 2.79·15-s + 16-s − 5.73·17-s − 4.79·18-s + 0.986·19-s − 20-s − 13.5·21-s + 4.34·22-s − 0.818·23-s + 2.79·24-s + 25-s − 5.01·27-s + 4.83·28-s + 5.85·29-s − 2.79·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s − 0.447·5-s + 1.13·6-s + 1.82·7-s − 0.353·8-s + 1.59·9-s + 0.316·10-s − 1.31·11-s − 0.806·12-s − 1.29·14-s + 0.720·15-s + 0.250·16-s − 1.39·17-s − 1.13·18-s + 0.226·19-s − 0.223·20-s − 2.94·21-s + 0.927·22-s − 0.170·23-s + 0.569·24-s + 0.200·25-s − 0.965·27-s + 0.914·28-s + 1.08·29-s − 0.509·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 - 0.986T + 19T^{2} \) |
| 23 | \( 1 + 0.818T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 + 0.820T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 + 7.91T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + 2.56T + 73T^{2} \) |
| 79 | \( 1 + 1.21T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.683720141944258819527457544138, −8.203544683708437054441479965966, −7.35048392353532763393904598243, −6.67127993012309958339380895671, −5.56729339157078440565901617758, −4.96594913099781001906058852406, −4.33107603947890978687671697195, −2.45815296905874907753326898761, −1.26947593626172361088308086612, 0,
1.26947593626172361088308086612, 2.45815296905874907753326898761, 4.33107603947890978687671697195, 4.96594913099781001906058852406, 5.56729339157078440565901617758, 6.67127993012309958339380895671, 7.35048392353532763393904598243, 8.203544683708437054441479965966, 8.683720141944258819527457544138
