L(s) = 1 | − 0.431·2-s + 3.18·3-s − 1.81·4-s − 5-s − 1.37·6-s − 7-s + 1.64·8-s + 7.15·9-s + 0.431·10-s + 3.80·11-s − 5.78·12-s − 2.66·13-s + 0.431·14-s − 3.18·15-s + 2.91·16-s + 4.14·17-s − 3.08·18-s + 0.188·19-s + 1.81·20-s − 3.18·21-s − 1.64·22-s − 5.11·23-s + 5.24·24-s + 25-s + 1.14·26-s + 13.2·27-s + 1.81·28-s + ⋯ |
L(s) = 1 | − 0.304·2-s + 1.84·3-s − 0.906·4-s − 0.447·5-s − 0.561·6-s − 0.377·7-s + 0.581·8-s + 2.38·9-s + 0.136·10-s + 1.14·11-s − 1.66·12-s − 0.737·13-s + 0.115·14-s − 0.822·15-s + 0.729·16-s + 1.00·17-s − 0.727·18-s + 0.0432·19-s + 0.405·20-s − 0.695·21-s − 0.349·22-s − 1.06·23-s + 1.07·24-s + 0.200·25-s + 0.225·26-s + 2.55·27-s + 0.342·28-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
\(L(1)\) |
\(\approx\) |
\(2.830494872\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.830494872\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.431T + 2T^{2} \) |
| 3 | \( 1 - 3.18T + 3T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 0.188T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 + 0.591T + 47T^{2} \) |
| 53 | \( 1 + 4.28T + 53T^{2} \) |
| 59 | \( 1 - 8.59T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 7.44T + 71T^{2} \) |
| 73 | \( 1 - 2.33T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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
−7.87047649607883129875799059778, −7.56241481955241968777504956649, −6.78080579099902082731505880854, −5.75597593189770451594215798623, −4.60036650464214973622397067061, −4.11526963858887196448197985432, −3.52252108800731269422506070343, −2.83453333908905959206013743654, −1.79456752755153655623899245737, −0.838848795876563996131958940493,
0.838848795876563996131958940493, 1.79456752755153655623899245737, 2.83453333908905959206013743654, 3.52252108800731269422506070343, 4.11526963858887196448197985432, 4.60036650464214973622397067061, 5.75597593189770451594215798623, 6.78080579099902082731505880854, 7.56241481955241968777504956649, 7.87047649607883129875799059778