L(s) = 1 | − 3·3-s + 2·5-s + 9·9-s + 52·11-s − 86·13-s − 6·15-s + 30·17-s + 4·19-s + 120·23-s − 121·25-s − 27·27-s + 246·29-s − 80·31-s − 156·33-s − 290·37-s + 258·39-s + 374·41-s + 164·43-s + 18·45-s − 464·47-s − 90·51-s − 162·53-s + 104·55-s − 12·57-s − 180·59-s + 666·61-s − 172·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.178·5-s + 1/3·9-s + 1.42·11-s − 1.83·13-s − 0.103·15-s + 0.428·17-s + 0.0482·19-s + 1.08·23-s − 0.967·25-s − 0.192·27-s + 1.57·29-s − 0.463·31-s − 0.822·33-s − 1.28·37-s + 1.05·39-s + 1.42·41-s + 0.581·43-s + 0.0596·45-s − 1.44·47-s − 0.247·51-s − 0.419·53-s + 0.254·55-s − 0.0278·57-s − 0.397·59-s + 1.39·61-s − 0.328·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.685063472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.685063472\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 290 T + p^{3} T^{2} \) |
| 41 | \( 1 - 374 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 666 T + p^{3} T^{2} \) |
| 67 | \( 1 + 628 T + p^{3} T^{2} \) |
| 71 | \( 1 - 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 518 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 220 T + p^{3} T^{2} \) |
| 89 | \( 1 - 774 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1086 T + p^{3} T^{2} \) |
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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
−9.581314322238467751977596734206, −8.710798272615979097284696317807, −7.52188394770121776334464710882, −6.91093841687542936263557881713, −6.07231271483134247264654808573, −5.08769409136982907798347048281, −4.37348089787001610266432809008, −3.16572961270463699423306988500, −1.89711002148195646288562658667, −0.69612231896776233797669022848,
0.69612231896776233797669022848, 1.89711002148195646288562658667, 3.16572961270463699423306988500, 4.37348089787001610266432809008, 5.08769409136982907798347048281, 6.07231271483134247264654808573, 6.91093841687542936263557881713, 7.52188394770121776334464710882, 8.710798272615979097284696317807, 9.581314322238467751977596734206