| L(s) = 1 | − 35.0·2-s + 118.·3-s + 718.·4-s + 1.63e3·5-s − 4.15e3·6-s − 1.07e4·7-s − 7.23e3·8-s − 5.62e3·9-s − 5.73e4·10-s + 2.11e4·11-s + 8.51e4·12-s − 7.10e4·13-s + 3.78e5·14-s + 1.93e5·15-s − 1.13e5·16-s + 1.97e5·18-s − 2.39e5·19-s + 1.17e6·20-s − 1.27e6·21-s − 7.43e5·22-s − 2.36e6·23-s − 8.58e5·24-s + 7.21e5·25-s + 2.49e6·26-s − 3.00e6·27-s − 7.74e6·28-s − 1.37e6·29-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.845·3-s + 1.40·4-s + 1.17·5-s − 1.31·6-s − 1.69·7-s − 0.624·8-s − 0.285·9-s − 1.81·10-s + 0.436·11-s + 1.18·12-s − 0.689·13-s + 2.63·14-s + 0.989·15-s − 0.434·16-s + 0.442·18-s − 0.420·19-s + 1.64·20-s − 1.43·21-s − 0.676·22-s − 1.76·23-s − 0.527·24-s + 0.369·25-s + 1.06·26-s − 1.08·27-s − 2.38·28-s − 0.362·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7185595360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7185595360\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 35.0T + 512T^{2} \) |
| 3 | \( 1 - 118.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.63e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.07e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.10e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 2.39e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.36e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.04e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.22e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.00e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.14e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.65e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.59e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.65e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.47e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.77e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.72e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.89e8T + 7.60e17T^{2} \) |
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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
−9.705404912549589502909201819189, −9.485719249507127671339690453089, −8.670255613687915127753694585478, −7.60626033191211465739944053459, −6.57163072680522086562713805634, −5.81217153620202256736884523537, −3.75634784257677717373265418219, −2.50348262676936714131813889950, −1.95077488711554868403191269918, −0.42880471800268433412562200871,
0.42880471800268433412562200871, 1.95077488711554868403191269918, 2.50348262676936714131813889950, 3.75634784257677717373265418219, 5.81217153620202256736884523537, 6.57163072680522086562713805634, 7.60626033191211465739944053459, 8.670255613687915127753694585478, 9.485719249507127671339690453089, 9.705404912549589502909201819189
