| L(s) = 1 | + 2-s + 3-s + 4-s + 0.722·5-s + 6-s + 8-s + 9-s + 0.722·10-s + 1.13·11-s + 12-s + 1.27·13-s + 0.722·15-s + 16-s − 3.93·17-s + 18-s + 2·19-s + 0.722·20-s + 1.13·22-s − 23-s + 24-s − 4.47·25-s + 1.27·26-s + 27-s − 5.76·29-s + 0.722·30-s + 6.20·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.323·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.228·10-s + 0.341·11-s + 0.288·12-s + 0.354·13-s + 0.186·15-s + 0.250·16-s − 0.954·17-s + 0.235·18-s + 0.458·19-s + 0.161·20-s + 0.241·22-s − 0.208·23-s + 0.204·24-s − 0.895·25-s + 0.250·26-s + 0.192·27-s − 1.07·29-s + 0.131·30-s + 1.11·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.647648575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.647648575\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.722T + 5T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 - 3.01T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.290T + 73T^{2} \) |
| 79 | \( 1 - 0.290T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 - 0.0631T + 89T^{2} \) |
| 97 | \( 1 - 0.750T + 97T^{2} \) |
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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
−7.961230797986625324529313371991, −7.18052267091335719603328682097, −6.52874096344927913485370941094, −5.83596903469258606763950057453, −5.15567566533791267208171543070, −4.03834245033923570215542615508, −3.90191286145674244954038286533, −2.63585755395282678018770221494, −2.16493599201513966600051225527, −0.997677632990168216434250776389,
0.997677632990168216434250776389, 2.16493599201513966600051225527, 2.63585755395282678018770221494, 3.90191286145674244954038286533, 4.03834245033923570215542615508, 5.15567566533791267208171543070, 5.83596903469258606763950057453, 6.52874096344927913485370941094, 7.18052267091335719603328682097, 7.961230797986625324529313371991
