| L(s) = 1 | + 2-s − 3.14·3-s + 4-s + 5-s − 3.14·6-s + 7-s + 8-s + 6.91·9-s + 10-s − 3.14·12-s − 5.43·13-s + 14-s − 3.14·15-s + 16-s + 0.477·17-s + 6.91·18-s + 5.91·19-s + 20-s − 3.14·21-s − 2.65·23-s − 3.14·24-s + 25-s − 5.43·26-s − 12.3·27-s + 28-s + 6.56·29-s − 3.14·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.447·5-s − 1.28·6-s + 0.377·7-s + 0.353·8-s + 2.30·9-s + 0.316·10-s − 0.909·12-s − 1.50·13-s + 0.267·14-s − 0.813·15-s + 0.250·16-s + 0.115·17-s + 1.63·18-s + 1.35·19-s + 0.223·20-s − 0.687·21-s − 0.553·23-s − 0.642·24-s + 0.200·25-s − 1.06·26-s − 2.37·27-s + 0.188·28-s + 1.21·29-s − 0.574·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 - 0.477T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + 9.68T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 9.42T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 0.566T + 59T^{2} \) |
| 61 | \( 1 + 2.04T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 9.54T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.17000976915584794206936911604, −6.68099972255142647977653042162, −5.73412871584645390868738769951, −5.41471964014715489249757851130, −4.89094808876283128385410572679, −4.26137037331529387689004716571, −3.19425624484420317891396926597, −2.06798211995058781350165300317, −1.23465302267431994723338321951, 0,
1.23465302267431994723338321951, 2.06798211995058781350165300317, 3.19425624484420317891396926597, 4.26137037331529387689004716571, 4.89094808876283128385410572679, 5.41471964014715489249757851130, 5.73412871584645390868738769951, 6.68099972255142647977653042162, 7.17000976915584794206936911604
