| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 13-s + 16-s − 17-s + 6·22-s − 3·23-s − 26-s + 7·31-s − 32-s + 34-s − 2·37-s + 5·41-s + 6·43-s − 6·44-s + 3·46-s + 47-s + 52-s + 8·59-s + 6·61-s − 7·62-s + 64-s + 2·67-s − 68-s − 71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s − 0.242·17-s + 1.27·22-s − 0.625·23-s − 0.196·26-s + 1.25·31-s − 0.176·32-s + 0.171·34-s − 0.328·37-s + 0.780·41-s + 0.914·43-s − 0.904·44-s + 0.442·46-s + 0.145·47-s + 0.138·52-s + 1.04·59-s + 0.768·61-s − 0.889·62-s + 1/8·64-s + 0.244·67-s − 0.121·68-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.661068003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.661068003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.81841686261038, −12.18795696848750, −11.76473235417405, −11.23747563122321, −10.78676593401232, −10.34888951889415, −10.11268699753693, −9.512876178400084, −9.049253822033677, −8.446902937124544, −8.052387948892649, −7.823874198630370, −7.158481411624931, −6.780444012336441, −6.066037590841369, −5.719971067864823, −5.201541111045515, −4.593978186927992, −4.093012524901904, −3.313128641901133, −2.840552472301458, −2.247473974932227, −1.917839940392997, −0.8310972516647331, −0.5005160760557654,
0.5005160760557654, 0.8310972516647331, 1.917839940392997, 2.247473974932227, 2.840552472301458, 3.313128641901133, 4.093012524901904, 4.593978186927992, 5.201541111045515, 5.719971067864823, 6.066037590841369, 6.780444012336441, 7.158481411624931, 7.823874198630370, 8.052387948892649, 8.446902937124544, 9.049253822033677, 9.512876178400084, 10.11268699753693, 10.34888951889415, 10.78676593401232, 11.23747563122321, 11.76473235417405, 12.18795696848750, 12.81841686261038
