| L(s) = 1 | + 2-s + 2.63·3-s + 4-s − 1.09·5-s + 2.63·6-s − 3.28·7-s + 8-s + 3.93·9-s − 1.09·10-s − 11-s + 2.63·12-s − 3.64·13-s − 3.28·14-s − 2.89·15-s + 16-s − 1.31·17-s + 3.93·18-s − 6.31·19-s − 1.09·20-s − 8.66·21-s − 22-s + 2.69·23-s + 2.63·24-s − 3.79·25-s − 3.64·26-s + 2.46·27-s − 3.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.491·5-s + 1.07·6-s − 1.24·7-s + 0.353·8-s + 1.31·9-s − 0.347·10-s − 0.301·11-s + 0.760·12-s − 1.01·13-s − 0.878·14-s − 0.747·15-s + 0.250·16-s − 0.319·17-s + 0.927·18-s − 1.44·19-s − 0.245·20-s − 1.88·21-s − 0.213·22-s + 0.561·23-s + 0.537·24-s − 0.758·25-s − 0.715·26-s + 0.473·27-s − 0.621·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 + 8.96T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 - 5.40T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 + 5.72T + 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 8.61T + 79T^{2} \) |
| 83 | \( 1 - 8.46T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.118006140928415430516392804716, −7.13544930126760904761772235685, −6.83196242970680247337783408486, −5.80553493108854339193100232097, −4.77301387035096165345382551942, −3.93045322321865435466803610282, −3.42487875302226977406166915909, −2.61249040060680466117642819958, −2.02527259697635081265115911866, 0,
2.02527259697635081265115911866, 2.61249040060680466117642819958, 3.42487875302226977406166915909, 3.93045322321865435466803610282, 4.77301387035096165345382551942, 5.80553493108854339193100232097, 6.83196242970680247337783408486, 7.13544930126760904761772235685, 8.118006140928415430516392804716
