| L(s) = 1 | − 2.32·2-s + 3.42·4-s + 0.317·5-s − 1.41·7-s − 3.31·8-s − 0.738·10-s + 4.19·13-s + 3.30·14-s + 0.876·16-s + 6.24·17-s + 3.09·19-s + 1.08·20-s + 4.52·23-s − 4.89·25-s − 9.77·26-s − 4.85·28-s − 2.92·29-s − 9.54·31-s + 4.59·32-s − 14.5·34-s − 0.449·35-s + 0.847·37-s − 7.21·38-s − 1.05·40-s + 3.86·41-s + 3.55·43-s − 10.5·46-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.71·4-s + 0.141·5-s − 0.536·7-s − 1.17·8-s − 0.233·10-s + 1.16·13-s + 0.882·14-s + 0.219·16-s + 1.51·17-s + 0.710·19-s + 0.242·20-s + 0.942·23-s − 0.979·25-s − 1.91·26-s − 0.917·28-s − 0.543·29-s − 1.71·31-s + 0.811·32-s − 2.49·34-s − 0.0760·35-s + 0.139·37-s − 1.17·38-s − 0.166·40-s + 0.604·41-s + 0.541·43-s − 1.55·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9783562418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9783562418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 - 0.317T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 0.847T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 3.38T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 2.69T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−7.59680419810872832383880897229, −7.41042691438381297626739612943, −6.49558791827811894065621993750, −5.83829342472017578344368716413, −5.22799327713644145331081967484, −3.83294313864035008779829085313, −3.32508312068441920187807910315, −2.28255686910910173717290947496, −1.37266274661149547159696049969, −0.66937351811870951305328837841,
0.66937351811870951305328837841, 1.37266274661149547159696049969, 2.28255686910910173717290947496, 3.32508312068441920187807910315, 3.83294313864035008779829085313, 5.22799327713644145331081967484, 5.83829342472017578344368716413, 6.49558791827811894065621993750, 7.41042691438381297626739612943, 7.59680419810872832383880897229
