| L(s) = 1 | + 2-s − 3-s + 4-s − 0.692·5-s − 6-s − 7-s + 8-s + 9-s − 0.692·10-s − 0.664·11-s − 12-s − 14-s + 0.692·15-s + 16-s − 7.74·17-s + 18-s − 3.89·19-s − 0.692·20-s + 21-s − 0.664·22-s + 0.841·23-s − 24-s − 4.52·25-s − 27-s − 28-s + 8.98·29-s + 0.692·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.309·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.218·10-s − 0.200·11-s − 0.288·12-s − 0.267·14-s + 0.178·15-s + 0.250·16-s − 1.87·17-s + 0.235·18-s − 0.894·19-s − 0.154·20-s + 0.218·21-s − 0.141·22-s + 0.175·23-s − 0.204·24-s − 0.904·25-s − 0.192·27-s − 0.188·28-s + 1.66·29-s + 0.126·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.763667464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.763667464\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.692T + 5T^{2} \) |
| 11 | \( 1 + 0.664T + 11T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 - 0.841T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 6.05T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 - 9.93T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 - 0.454T + 89T^{2} \) |
| 97 | \( 1 - 8.61T + 97T^{2} \) |
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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
−7.931513506684801469721040146660, −6.86119405662943590164849080497, −6.46625138155622114803946481056, −6.00681172260235408924471831463, −4.79789658553313099990757923798, −4.58592351449154998824457586394, −3.74823239854978390678139938892, −2.74315647899029034482060005184, −2.01451710525541718874552850798, −0.59910144327633140994847215715,
0.59910144327633140994847215715, 2.01451710525541718874552850798, 2.74315647899029034482060005184, 3.74823239854978390678139938892, 4.58592351449154998824457586394, 4.79789658553313099990757923798, 6.00681172260235408924471831463, 6.46625138155622114803946481056, 6.86119405662943590164849080497, 7.931513506684801469721040146660
