| L(s) = 1 | − 2-s + 4-s + 0.585·5-s − 8-s − 0.585·10-s − 11-s − 3.82·13-s + 16-s − 3.65·17-s − 0.585·19-s + 0.585·20-s + 22-s + 6.24·23-s − 4.65·25-s + 3.82·26-s − 2.65·29-s − 4·31-s − 32-s + 3.65·34-s − 9.41·37-s + 0.585·38-s − 0.585·40-s + 5.41·41-s − 5.65·43-s − 44-s − 6.24·46-s + 10.4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.261·5-s − 0.353·8-s − 0.185·10-s − 0.301·11-s − 1.06·13-s + 0.250·16-s − 0.886·17-s − 0.134·19-s + 0.130·20-s + 0.213·22-s + 1.30·23-s − 0.931·25-s + 0.750·26-s − 0.493·29-s − 0.718·31-s − 0.176·32-s + 0.627·34-s − 1.54·37-s + 0.0950·38-s − 0.0926·40-s + 0.845·41-s − 0.862·43-s − 0.150·44-s − 0.920·46-s + 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9414401584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9414401584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 0.585T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 9.41T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−7.51183389471990926566209118894, −7.19576411792399136382220659439, −6.51882453803484115696885308398, −5.60378833607157166717260971034, −5.09245245511981730746161516684, −4.18720382324560450632702719503, −3.25167360235717898908621419292, −2.36802872553764179731784778409, −1.80070410638963361008062689809, −0.49362482980919079971154421053,
0.49362482980919079971154421053, 1.80070410638963361008062689809, 2.36802872553764179731784778409, 3.25167360235717898908621419292, 4.18720382324560450632702719503, 5.09245245511981730746161516684, 5.60378833607157166717260971034, 6.51882453803484115696885308398, 7.19576411792399136382220659439, 7.51183389471990926566209118894
