L(s) = 1 | + 9·3-s + 64.5·5-s + 11.9·7-s + 81·9-s − 303.·11-s − 771.·13-s + 580.·15-s + 401.·17-s + 273.·19-s + 107.·21-s + 529·23-s + 1.03e3·25-s + 729·27-s − 5.60e3·29-s + 1.95e3·31-s − 2.72e3·33-s + 773.·35-s − 6.65e3·37-s − 6.94e3·39-s + 1.35e4·41-s + 2.08e4·43-s + 5.22e3·45-s − 2.76e4·47-s − 1.66e4·49-s + 3.61e3·51-s + 1.62e4·53-s − 1.95e4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.15·5-s + 0.0925·7-s + 0.333·9-s − 0.755·11-s − 1.26·13-s + 0.666·15-s + 0.337·17-s + 0.173·19-s + 0.0534·21-s + 0.208·23-s + 0.331·25-s + 0.192·27-s − 1.23·29-s + 0.365·31-s − 0.436·33-s + 0.106·35-s − 0.799·37-s − 0.730·39-s + 1.26·41-s + 1.72·43-s + 0.384·45-s − 1.82·47-s − 0.991·49-s + 0.194·51-s + 0.796·53-s − 0.871·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
good | 5 | \( 1 - 64.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 11.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 303.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 771.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 401.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 273.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.76e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.62e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.86e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.59e4T + 8.58e9T^{2} \) |
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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
−8.882793339101178273027953299021, −7.76667911768378989815979858920, −7.26191260520026033913321052235, −6.05674790204473024430746462868, −5.33516258584607300208641971483, −4.45499840717928409262856213269, −3.08290593860206336367915532526, −2.34817781075865006948300196469, −1.48594867335531457030862585710, 0,
1.48594867335531457030862585710, 2.34817781075865006948300196469, 3.08290593860206336367915532526, 4.45499840717928409262856213269, 5.33516258584607300208641971483, 6.05674790204473024430746462868, 7.26191260520026033913321052235, 7.76667911768378989815979858920, 8.882793339101178273027953299021