| L(s) = 1 | + 0.295·2-s − 2.62·3-s − 1.91·4-s − 5-s − 0.775·6-s − 7-s − 1.15·8-s + 3.89·9-s − 0.295·10-s + 5.02·12-s − 1.88·13-s − 0.295·14-s + 2.62·15-s + 3.48·16-s − 5.47·17-s + 1.15·18-s − 2.12·19-s + 1.91·20-s + 2.62·21-s + 3.16·23-s + 3.03·24-s + 25-s − 0.558·26-s − 2.35·27-s + 1.91·28-s − 8.99·29-s + 0.775·30-s + ⋯ |
| L(s) = 1 | + 0.208·2-s − 1.51·3-s − 0.956·4-s − 0.447·5-s − 0.316·6-s − 0.377·7-s − 0.408·8-s + 1.29·9-s − 0.0934·10-s + 1.45·12-s − 0.523·13-s − 0.0789·14-s + 0.678·15-s + 0.871·16-s − 1.32·17-s + 0.271·18-s − 0.486·19-s + 0.427·20-s + 0.573·21-s + 0.659·23-s + 0.619·24-s + 0.200·25-s − 0.109·26-s − 0.453·27-s + 0.361·28-s − 1.67·29-s + 0.141·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.007875721724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007875721724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.295T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 - 1.56T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 1.56T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.642301210514692098864514252089, −7.34681312628344485292964927886, −6.91816960708396772547374957304, −6.00327679800949241035746893901, −5.36946391613539348725042916852, −4.76143302916040798173018777786, −4.09922378758839933114382578293, −3.22876816560349100578907922694, −1.68614642625041103361336682237, −0.05214561352375145175885280706,
0.05214561352375145175885280706, 1.68614642625041103361336682237, 3.22876816560349100578907922694, 4.09922378758839933114382578293, 4.76143302916040798173018777786, 5.36946391613539348725042916852, 6.00327679800949241035746893901, 6.91816960708396772547374957304, 7.34681312628344485292964927886, 8.642301210514692098864514252089
