| L(s) = 1 | − 2-s + 1.80·3-s + 4-s + 5-s − 1.80·6-s − 1.55·7-s − 8-s + 0.246·9-s − 10-s − 4.49·11-s + 1.80·12-s + 1.55·14-s + 1.80·15-s + 16-s + 0.396·17-s − 0.246·18-s − 5.60·19-s + 20-s − 2.80·21-s + 4.49·22-s − 3.02·23-s − 1.80·24-s + 25-s − 4.96·27-s − 1.55·28-s − 0.692·29-s − 1.80·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.447·5-s − 0.735·6-s − 0.587·7-s − 0.353·8-s + 0.0823·9-s − 0.316·10-s − 1.35·11-s + 0.520·12-s + 0.415·14-s + 0.465·15-s + 0.250·16-s + 0.0960·17-s − 0.0582·18-s − 1.28·19-s + 0.223·20-s − 0.611·21-s + 0.958·22-s − 0.631·23-s − 0.367·24-s + 0.200·25-s − 0.954·27-s − 0.293·28-s − 0.128·29-s − 0.328·30-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 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 17 | \( 1 - 0.396T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 0.692T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 5.16T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.951957386054073538911169039387, −8.155932114934338152166094577584, −7.77574267136148670660816568122, −6.62005487638181958211139928004, −5.93891504604407512413047824106, −4.80861270103080427272032062753, −3.46436727402739108592896516717, −2.67509984951510738860333429198, −1.92798746696859126923452526842, 0,
1.92798746696859126923452526842, 2.67509984951510738860333429198, 3.46436727402739108592896516717, 4.80861270103080427272032062753, 5.93891504604407512413047824106, 6.62005487638181958211139928004, 7.77574267136148670660816568122, 8.155932114934338152166094577584, 8.951957386054073538911169039387
