| L(s) = 1 | − 0.449·2-s + 1.97·3-s − 1.79·4-s − 2.37·5-s − 0.887·6-s − 0.236·7-s + 1.70·8-s + 0.892·9-s + 1.06·10-s + 11-s − 3.54·12-s − 0.980·13-s + 0.106·14-s − 4.69·15-s + 2.82·16-s + 17-s − 0.401·18-s − 3.74·19-s + 4.27·20-s − 0.467·21-s − 0.449·22-s − 6.39·23-s + 3.37·24-s + 0.651·25-s + 0.441·26-s − 4.15·27-s + 0.425·28-s + ⋯ |
| L(s) = 1 | − 0.318·2-s + 1.13·3-s − 0.898·4-s − 1.06·5-s − 0.362·6-s − 0.0895·7-s + 0.603·8-s + 0.297·9-s + 0.338·10-s + 0.301·11-s − 1.02·12-s − 0.271·13-s + 0.0284·14-s − 1.21·15-s + 0.706·16-s + 0.242·17-s − 0.0946·18-s − 0.859·19-s + 0.955·20-s − 0.101·21-s − 0.0959·22-s − 1.33·23-s + 0.687·24-s + 0.130·25-s + 0.0865·26-s − 0.800·27-s + 0.0804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047187664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047187664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.449T + 2T^{2} \) |
| 3 | \( 1 - 1.97T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 13 | \( 1 + 0.980T + 13T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 + 0.645T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 - 8.73T + 41T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.83T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 + 1.37T + 83T^{2} \) |
| 89 | \( 1 - 3.78T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.86581884285959628691727583220, −7.73608076433338210512897510878, −6.57518243017753395694078137886, −5.77483985412143987406185176664, −4.67243062838739999245401033887, −4.14173051110912394239382875296, −3.62213676965333629550607038078, −2.79160336828598752069597803734, −1.80082430947891189623015469967, −0.49689535352824754508801367457,
0.49689535352824754508801367457, 1.80082430947891189623015469967, 2.79160336828598752069597803734, 3.62213676965333629550607038078, 4.14173051110912394239382875296, 4.67243062838739999245401033887, 5.77483985412143987406185176664, 6.57518243017753395694078137886, 7.73608076433338210512897510878, 7.86581884285959628691727583220
