L(s) = 1 | + 2-s − 3-s + 4-s − 0.879·5-s − 6-s + 8-s + 9-s − 0.879·10-s − 4.07·11-s − 12-s + 2.22·13-s + 0.879·15-s + 16-s + 7.75·17-s + 18-s + 0.585·19-s − 0.879·20-s − 4.07·22-s − 23-s − 24-s − 4.22·25-s + 2.22·26-s − 27-s + 4.60·29-s + 0.879·30-s + 0.585·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.393·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.278·10-s − 1.22·11-s − 0.288·12-s + 0.616·13-s + 0.227·15-s + 0.250·16-s + 1.88·17-s + 0.235·18-s + 0.134·19-s − 0.196·20-s − 0.868·22-s − 0.208·23-s − 0.204·24-s − 0.845·25-s + 0.435·26-s − 0.192·27-s + 0.855·29-s + 0.160·30-s + 0.105·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.300409592\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.300409592\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 0.879T + 5T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 0.585T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 + 0.124T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 - 0.0515T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 - 4.07T + 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 6.99T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.85497501162734747474169065357, −7.32449970129932186659500524627, −6.30667186432654293979231144948, −5.82213310589560271191843626403, −5.14175490160518899970121119110, −4.52984011965150928870617580493, −3.54437812802650912824602900762, −3.03078201733257928878567711719, −1.85111588495953664357878254988, −0.72359501173465975687925293271,
0.72359501173465975687925293271, 1.85111588495953664357878254988, 3.03078201733257928878567711719, 3.54437812802650912824602900762, 4.52984011965150928870617580493, 5.14175490160518899970121119110, 5.82213310589560271191843626403, 6.30667186432654293979231144948, 7.32449970129932186659500524627, 7.85497501162734747474169065357