L(s) = 1 | + 2-s − 0.869·3-s + 4-s − 5-s − 0.869·6-s − 1.24·7-s + 8-s − 2.24·9-s − 10-s − 1.88·11-s − 0.869·12-s − 3.05·13-s − 1.24·14-s + 0.869·15-s + 16-s − 5.93·17-s − 2.24·18-s + 6.20·19-s − 20-s + 1.08·21-s − 1.88·22-s − 2.62·23-s − 0.869·24-s + 25-s − 3.05·26-s + 4.56·27-s − 1.24·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.502·3-s + 0.5·4-s − 0.447·5-s − 0.355·6-s − 0.469·7-s + 0.353·8-s − 0.747·9-s − 0.316·10-s − 0.569·11-s − 0.251·12-s − 0.847·13-s − 0.332·14-s + 0.224·15-s + 0.250·16-s − 1.44·17-s − 0.528·18-s + 1.42·19-s − 0.223·20-s + 0.235·21-s − 0.402·22-s − 0.546·23-s − 0.177·24-s + 0.200·25-s − 0.598·26-s + 0.877·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.254366079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254366079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 0.869T + 3T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 + 0.0894T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 0.163T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 - 4.69T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 79 | \( 1 + 8.64T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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
−7.80859926833702252173958991357, −7.31981557648179780233042529452, −6.52277934952702050533830165756, −5.88080243179027509024574552623, −5.14236319333751352913366301596, −4.62722538314161165428312798472, −3.63933698599938601658468945897, −2.89079913367894705357700119192, −2.14132579503960635860267152645, −0.50840621040337703530573099902,
0.50840621040337703530573099902, 2.14132579503960635860267152645, 2.89079913367894705357700119192, 3.63933698599938601658468945897, 4.62722538314161165428312798472, 5.14236319333751352913366301596, 5.88080243179027509024574552623, 6.52277934952702050533830165756, 7.31981557648179780233042529452, 7.80859926833702252173958991357