| L(s) = 1 | + 2-s + 4-s − 1.73·5-s + 8-s − 1.73·10-s + 1.73·13-s + 16-s + 3·17-s − 6.92·19-s − 1.73·20-s + 6.92·23-s − 2.00·25-s + 1.73·26-s + 9·29-s + 4·31-s + 32-s + 3·34-s + 7·37-s − 6.92·38-s − 1.73·40-s + 9·41-s + 6.92·46-s − 7·49-s − 2.00·50-s + 1.73·52-s − 5.19·53-s + 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.774·5-s + 0.353·8-s − 0.547·10-s + 0.480·13-s + 0.250·16-s + 0.727·17-s − 1.58·19-s − 0.387·20-s + 1.44·23-s − 0.400·25-s + 0.339·26-s + 1.67·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s + 1.15·37-s − 1.12·38-s − 0.273·40-s + 1.40·41-s + 1.02·46-s − 49-s − 0.282·50-s + 0.240·52-s − 0.713·53-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.520808527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520808527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−8.881618302401695426205318307940, −8.192740556546012022838364822134, −7.50988738366092904360731583852, −6.56155325509407443417368630095, −6.00354086107244130447988449614, −4.83711827042756013747316494399, −4.26831145603937878514727218893, −3.36069933875381399794068640967, −2.46887728442912172361828947908, −0.965957262074393882604932263689,
0.965957262074393882604932263689, 2.46887728442912172361828947908, 3.36069933875381399794068640967, 4.26831145603937878514727218893, 4.83711827042756013747316494399, 6.00354086107244130447988449614, 6.56155325509407443417368630095, 7.50988738366092904360731583852, 8.192740556546012022838364822134, 8.881618302401695426205318307940
