| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 9-s + 1.41·10-s + 16-s − 18-s − 1.41·20-s + 1.00·25-s + 1.41·29-s − 32-s + 36-s + 1.41·37-s + 1.41·40-s + 1.41·41-s − 1.41·45-s + 49-s − 1.00·50-s − 1.41·58-s − 1.41·61-s + 64-s − 72-s + 1.41·73-s − 1.41·74-s − 1.41·80-s + 81-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 9-s + 1.41·10-s + 16-s − 18-s − 1.41·20-s + 1.00·25-s + 1.41·29-s − 32-s + 36-s + 1.41·37-s + 1.41·40-s + 1.41·41-s − 1.41·45-s + 49-s − 1.00·50-s − 1.41·58-s − 1.41·61-s + 64-s − 72-s + 1.41·73-s − 1.41·74-s − 1.41·80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5729612795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5729612795\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.947616488911611667805149713777, −9.151671926075898747539385697983, −8.220372922176685241258245346748, −7.66396180224138514176893932937, −7.02312437043636480573330298589, −6.11209684634065673106790697179, −4.63891897612742910562639083406, −3.79318585213712881284557812590, −2.61488359518915618297368300662, −1.01712437104286264984381406238,
1.01712437104286264984381406238, 2.61488359518915618297368300662, 3.79318585213712881284557812590, 4.63891897612742910562639083406, 6.11209684634065673106790697179, 7.02312437043636480573330298589, 7.66396180224138514176893932937, 8.220372922176685241258245346748, 9.151671926075898747539385697983, 9.947616488911611667805149713777
