| L(s) = 1 | + 20.0·3-s + 33.3·5-s − 49·7-s + 159.·9-s − 660.·11-s + 145.·13-s + 668.·15-s + 435.·17-s − 1.52e3·19-s − 982.·21-s − 206.·23-s − 2.01e3·25-s − 1.68e3·27-s + 2.03e3·29-s + 1.15e3·31-s − 1.32e4·33-s − 1.63e3·35-s − 6.75e3·37-s + 2.92e3·39-s − 1.53e4·41-s − 6.25e3·43-s + 5.29e3·45-s − 1.26e4·47-s + 2.40e3·49-s + 8.72e3·51-s − 1.08e4·53-s − 2.20e4·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s + 0.596·5-s − 0.377·7-s + 0.654·9-s − 1.64·11-s + 0.239·13-s + 0.766·15-s + 0.365·17-s − 0.970·19-s − 0.486·21-s − 0.0814·23-s − 0.644·25-s − 0.444·27-s + 0.450·29-s + 0.215·31-s − 2.11·33-s − 0.225·35-s − 0.811·37-s + 0.307·39-s − 1.42·41-s − 0.516·43-s + 0.390·45-s − 0.832·47-s + 0.142·49-s + 0.469·51-s − 0.530·53-s − 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 20.0T + 243T^{2} \) |
| 5 | \( 1 - 33.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 660.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 145.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 435.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 206.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.53e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.98e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.73e4T + 8.58e9T^{2} \) |
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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
−10.01490794763774398046830199115, −8.730958984240987228674892061522, −8.253619335234944802828676320529, −7.27076225926904819156826996967, −6.06999873853488773201046201847, −5.02796578930623352348626099233, −3.61058168782483100650133293487, −2.69643381960503211420575835251, −1.86662823528514726484493683950, 0,
1.86662823528514726484493683950, 2.69643381960503211420575835251, 3.61058168782483100650133293487, 5.02796578930623352348626099233, 6.06999873853488773201046201847, 7.27076225926904819156826996967, 8.253619335234944802828676320529, 8.730958984240987228674892061522, 10.01490794763774398046830199115
