| L(s) = 1 | + 2.62·3-s − 11.5·5-s − 23.0·7-s − 20.1·9-s − 1.19·11-s + 58.3·13-s − 30.2·15-s − 138.·19-s − 60.4·21-s − 180.·23-s + 7.98·25-s − 123.·27-s + 115.·29-s − 210.·31-s − 3.13·33-s + 265.·35-s − 210.·37-s + 152.·39-s + 297.·41-s − 174.·43-s + 232.·45-s − 199.·47-s + 188.·49-s + 706.·53-s + 13.7·55-s − 364.·57-s + 182.·59-s + ⋯ |
| L(s) = 1 | + 0.504·3-s − 1.03·5-s − 1.24·7-s − 0.745·9-s − 0.0327·11-s + 1.24·13-s − 0.520·15-s − 1.67·19-s − 0.627·21-s − 1.64·23-s + 0.0638·25-s − 0.880·27-s + 0.736·29-s − 1.22·31-s − 0.0165·33-s + 1.28·35-s − 0.936·37-s + 0.628·39-s + 1.13·41-s − 0.619·43-s + 0.769·45-s − 0.619·47-s + 0.548·49-s + 1.83·53-s + 0.0337·55-s − 0.846·57-s + 0.403·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4543237608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4543237608\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.62T + 27T^{2} \) |
| 5 | \( 1 + 11.5T + 125T^{2} \) |
| 7 | \( 1 + 23.0T + 343T^{2} \) |
| 11 | \( 1 + 1.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 210.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 489.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 167.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 818.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 110.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 118.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 400.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 924.T + 9.12e5T^{2} \) |
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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
−8.590015063321111644721389586210, −8.086214642962766885544274372373, −7.16635923523267929728025705740, −6.23920855095234805458688491501, −5.80013527138334605731756371077, −4.24637665487955670445266885591, −3.73911830017301135566875998580, −3.03532095223062957497593793283, −1.94399288202759884286304490774, −0.27541894218100982592569703578,
0.27541894218100982592569703578, 1.94399288202759884286304490774, 3.03532095223062957497593793283, 3.73911830017301135566875998580, 4.24637665487955670445266885591, 5.80013527138334605731756371077, 6.23920855095234805458688491501, 7.16635923523267929728025705740, 8.086214642962766885544274372373, 8.590015063321111644721389586210
