L(s) = 1 | − 2-s + 3-s + 4-s + 2.73·5-s − 6-s + 7-s − 8-s + 9-s − 2.73·10-s + 1.73·11-s + 12-s − 14-s + 2.73·15-s + 16-s + 0.267·17-s − 18-s − 19-s + 2.73·20-s + 21-s − 1.73·22-s + 3.46·23-s − 24-s + 2.46·25-s + 27-s + 28-s − 0.464·29-s − 2.73·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.22·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.863·10-s + 0.522·11-s + 0.288·12-s − 0.267·14-s + 0.705·15-s + 0.250·16-s + 0.0649·17-s − 0.235·18-s − 0.229·19-s + 0.610·20-s + 0.218·21-s − 0.369·22-s + 0.722·23-s − 0.204·24-s + 0.492·25-s + 0.192·27-s + 0.188·28-s − 0.0861·29-s − 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.777144057\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.777144057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 17 | \( 1 - 0.267T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.956631668692628380685886451970, −7.41917788607948480205610337193, −6.48620827009335767183281825138, −6.11487593894164456618605309450, −5.15918797883213657015177948409, −4.38255638184367424323185062122, −3.29594091665378146016179835872, −2.48554426471032434516161669766, −1.77865485860307062731330305620, −0.972442254934560421103330310643,
0.972442254934560421103330310643, 1.77865485860307062731330305620, 2.48554426471032434516161669766, 3.29594091665378146016179835872, 4.38255638184367424323185062122, 5.15918797883213657015177948409, 6.11487593894164456618605309450, 6.48620827009335767183281825138, 7.41917788607948480205610337193, 7.956631668692628380685886451970