| L(s) = 1 | − 1.44·2-s − 0.334·3-s + 0.0995·4-s − 5-s + 0.484·6-s − 7-s + 2.75·8-s − 2.88·9-s + 1.44·10-s + 1.28·11-s − 0.0332·12-s − 2.76·13-s + 1.44·14-s + 0.334·15-s − 4.18·16-s − 5.48·17-s + 4.18·18-s + 2.60·19-s − 0.0995·20-s + 0.334·21-s − 1.85·22-s − 6.54·23-s − 0.920·24-s + 25-s + 4.00·26-s + 1.96·27-s − 0.0995·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 0.193·3-s + 0.0497·4-s − 0.447·5-s + 0.197·6-s − 0.377·7-s + 0.973·8-s − 0.962·9-s + 0.458·10-s + 0.386·11-s − 0.00960·12-s − 0.767·13-s + 0.387·14-s + 0.0863·15-s − 1.04·16-s − 1.32·17-s + 0.986·18-s + 0.598·19-s − 0.0222·20-s + 0.0729·21-s − 0.395·22-s − 1.36·23-s − 0.187·24-s + 0.200·25-s + 0.786·26-s + 0.378·27-s − 0.0188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.004099772483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004099772483\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 + 0.334T + 3T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 - 4.75T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 8.15T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
| 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
−7.83213410021304468834404425487, −7.38600608250343119783928856124, −6.65622444209597698418959238238, −5.80020135936654710601753427044, −5.08853101423880832149228595392, −4.22105762668406118920674286899, −3.56459370015270261777198370997, −2.44989801935416905570088920453, −1.60526495195135625166270919008, −0.03683445374973232889818076859,
0.03683445374973232889818076859, 1.60526495195135625166270919008, 2.44989801935416905570088920453, 3.56459370015270261777198370997, 4.22105762668406118920674286899, 5.08853101423880832149228595392, 5.80020135936654710601753427044, 6.65622444209597698418959238238, 7.38600608250343119783928856124, 7.83213410021304468834404425487
