L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s − 2.60·13-s + 14-s − 15-s + 16-s + 4.60·17-s − 18-s − 2.60·19-s − 20-s − 21-s + 2·22-s + 23-s − 24-s + 25-s + 2.60·26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.722·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 1.11·17-s − 0.235·18-s − 0.597·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.510·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 1.39T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.039919680204466614987862539193, −7.32351862606737712027502723467, −6.80078597075796584936728177354, −5.81654082660843496525887630050, −4.95296779091825928960110423220, −4.01770057736007391864931177361, −3.05916569537052200155536694523, −2.51678768453414848261967850860, −1.28565551540535630306896476209, 0,
1.28565551540535630306896476209, 2.51678768453414848261967850860, 3.05916569537052200155536694523, 4.01770057736007391864931177361, 4.95296779091825928960110423220, 5.81654082660843496525887630050, 6.80078597075796584936728177354, 7.32351862606737712027502723467, 8.039919680204466614987862539193