| L(s) = 1 | + 2-s + 0.714·3-s + 4-s − 0.558·5-s + 0.714·6-s − 7-s + 8-s − 2.48·9-s − 0.558·10-s + 0.714·12-s − 4.53·13-s − 14-s − 0.399·15-s + 16-s − 6.72·17-s − 2.48·18-s − 2.82·19-s − 0.558·20-s − 0.714·21-s − 5.77·23-s + 0.714·24-s − 4.68·25-s − 4.53·26-s − 3.92·27-s − 28-s + 6.24·29-s − 0.399·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.412·3-s + 0.5·4-s − 0.249·5-s + 0.291·6-s − 0.377·7-s + 0.353·8-s − 0.829·9-s − 0.176·10-s + 0.206·12-s − 1.25·13-s − 0.267·14-s − 0.103·15-s + 0.250·16-s − 1.63·17-s − 0.586·18-s − 0.647·19-s − 0.124·20-s − 0.155·21-s − 1.20·23-s + 0.145·24-s − 0.937·25-s − 0.890·26-s − 0.755·27-s − 0.188·28-s + 1.15·29-s − 0.0728·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.714T + 3T^{2} \) |
| 5 | \( 1 + 0.558T + 5T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 0.188T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 3.40T + 83T^{2} \) |
| 89 | \( 1 + 8.40T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−8.853356522310974200869648498173, −8.133481579231814140122083657474, −7.30837493072117756410485196692, −6.41178490597023864898829222565, −5.75148956427079458172030187530, −4.57766453837086731530700562506, −4.01796709914219466864026197833, −2.73568692115974159230597588416, −2.24920365858829096182154264091, 0,
2.24920365858829096182154264091, 2.73568692115974159230597588416, 4.01796709914219466864026197833, 4.57766453837086731530700562506, 5.75148956427079458172030187530, 6.41178490597023864898829222565, 7.30837493072117756410485196692, 8.133481579231814140122083657474, 8.853356522310974200869648498173
