| L(s) = 1 | − 0.246·2-s − 1.93·4-s + 0.972·8-s − 3·9-s − 6.63·11-s + 3.63·16-s + 0.740·18-s + 1.63·22-s − 6.93·23-s − 5·25-s − 6.39·29-s − 2.84·32-s + 5.81·36-s − 11.6·37-s + 8.51·43-s + 12.8·44-s + 1.71·46-s + 1.23·50-s − 8.09·53-s + 1.57·58-s − 6.57·64-s − 14.5·67-s + 8.71·71-s − 2.91·72-s + 2.87·74-s − 7.42·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.174·2-s − 0.969·4-s + 0.343·8-s − 9-s − 1.99·11-s + 0.909·16-s + 0.174·18-s + 0.349·22-s − 1.44·23-s − 25-s − 1.18·29-s − 0.502·32-s + 0.969·36-s − 1.91·37-s + 1.29·43-s + 1.93·44-s + 0.252·46-s + 0.174·50-s − 1.11·53-s + 0.207·58-s − 0.821·64-s − 1.78·67-s + 1.03·71-s − 0.343·72-s + 0.334·74-s − 0.835·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08688827003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08688827003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.246T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.63T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8.51T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 8.71T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.81357463275353095844970788024, −7.54306975963146807091590421453, −6.21673723382262336310841587809, −5.47463945051036417987000066542, −5.28350774152048394399426726642, −4.26956364290653778556671545272, −3.52421891911014343964421715296, −2.67733287579765889486480497030, −1.80024561661407616511628142460, −0.14160764955539280153365516009,
0.14160764955539280153365516009, 1.80024561661407616511628142460, 2.67733287579765889486480497030, 3.52421891911014343964421715296, 4.26956364290653778556671545272, 5.28350774152048394399426726642, 5.47463945051036417987000066542, 6.21673723382262336310841587809, 7.54306975963146807091590421453, 7.81357463275353095844970788024
