| L(s) = 1 | − 2.56·2-s − 3-s + 4.57·4-s + 5-s + 2.56·6-s − 6.59·8-s + 9-s − 2.56·10-s − 11-s − 4.57·12-s + 1.89·13-s − 15-s + 7.76·16-s − 7.14·17-s − 2.56·18-s − 5.78·19-s + 4.57·20-s + 2.56·22-s + 1.58·23-s + 6.59·24-s + 25-s − 4.86·26-s − 27-s − 5.41·29-s + 2.56·30-s − 2.70·31-s − 6.71·32-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s + 0.447·5-s + 1.04·6-s − 2.33·8-s + 0.333·9-s − 0.810·10-s − 0.301·11-s − 1.31·12-s + 0.526·13-s − 0.258·15-s + 1.94·16-s − 1.73·17-s − 0.604·18-s − 1.32·19-s + 1.02·20-s + 0.546·22-s + 0.331·23-s + 1.34·24-s + 0.200·25-s − 0.954·26-s − 0.192·27-s − 1.00·29-s + 0.468·30-s − 0.485·31-s − 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 13 | \( 1 - 1.89T + 13T^{2} \) |
| 17 | \( 1 + 7.14T + 17T^{2} \) |
| 19 | \( 1 + 5.78T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 2.70T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 - 7.36T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 6.06T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.49298638725072324141018599714, −6.98125242579648079313866777284, −6.23856306687509498311975932319, −5.87977563296946676090031089694, −4.73545845908509677087347558805, −3.85250234634842480490002692241, −2.42704385455916637894458428391, −2.08607146680944739290333775353, −0.963279967985520266073701846021, 0,
0.963279967985520266073701846021, 2.08607146680944739290333775353, 2.42704385455916637894458428391, 3.85250234634842480490002692241, 4.73545845908509677087347558805, 5.87977563296946676090031089694, 6.23856306687509498311975932319, 6.98125242579648079313866777284, 7.49298638725072324141018599714
