L(s) = 1 | − 2-s + 3.29·3-s + 4-s + 1.67·5-s − 3.29·6-s − 4.86·7-s − 8-s + 7.85·9-s − 1.67·10-s − 11-s + 3.29·12-s − 3.50·13-s + 4.86·14-s + 5.52·15-s + 16-s + 3.62·17-s − 7.85·18-s + 1.67·20-s − 16.0·21-s + 22-s + 2.18·23-s − 3.29·24-s − 2.18·25-s + 3.50·26-s + 15.9·27-s − 4.86·28-s + 4.29·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.749·5-s − 1.34·6-s − 1.84·7-s − 0.353·8-s + 2.61·9-s − 0.530·10-s − 0.301·11-s + 0.951·12-s − 0.971·13-s + 1.30·14-s + 1.42·15-s + 0.250·16-s + 0.879·17-s − 1.85·18-s + 0.374·20-s − 3.50·21-s + 0.213·22-s + 0.455·23-s − 0.672·24-s − 0.437·25-s + 0.687·26-s + 3.07·27-s − 0.920·28-s + 0.796·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.924635756\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.924635756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.63T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 8.91T + 97T^{2} \) |
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
−7.79165888709767037762945332646, −7.44657803145929372896986909980, −6.63435070446240096512276107672, −6.13221201748980863557964409493, −4.97701788813054301175973032767, −3.90379183772987633181487196184, −3.08351047984136269126662976171, −2.74183010101730594632717439814, −2.04887990584138918442774850554, −0.844550390328305882231882763188,
0.844550390328305882231882763188, 2.04887990584138918442774850554, 2.74183010101730594632717439814, 3.08351047984136269126662976171, 3.90379183772987633181487196184, 4.97701788813054301175973032767, 6.13221201748980863557964409493, 6.63435070446240096512276107672, 7.44657803145929372896986909980, 7.79165888709767037762945332646