L(s) = 1 | − 2-s − 2.30·3-s + 4-s + 0.374·5-s + 2.30·6-s − 8-s + 2.32·9-s − 0.374·10-s + 2.34·11-s − 2.30·12-s − 0.133·13-s − 0.864·15-s + 16-s − 1.70·17-s − 2.32·18-s − 0.725·19-s + 0.374·20-s − 2.34·22-s + 0.678·23-s + 2.30·24-s − 4.85·25-s + 0.133·26-s + 1.54·27-s + 2.81·29-s + 0.864·30-s − 2.53·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.33·3-s + 0.5·4-s + 0.167·5-s + 0.942·6-s − 0.353·8-s + 0.776·9-s − 0.118·10-s + 0.706·11-s − 0.666·12-s − 0.0370·13-s − 0.223·15-s + 0.250·16-s − 0.413·17-s − 0.549·18-s − 0.166·19-s + 0.0837·20-s − 0.499·22-s + 0.141·23-s + 0.471·24-s − 0.971·25-s + 0.0262·26-s + 0.297·27-s + 0.523·29-s + 0.157·30-s − 0.455·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6981523686\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6981523686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 0.374T + 5T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 0.133T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 0.725T + 19T^{2} \) |
| 23 | \( 1 - 0.678T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 - 3.99T + 47T^{2} \) |
| 53 | \( 1 + 1.22T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 1.22T + 61T^{2} \) |
| 67 | \( 1 - 0.682T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
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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
−8.500444170214735020367757019785, −7.62521798233693919309900186987, −6.91044421897281929756550200293, −6.14458410012205572738532738795, −5.82435646862473482612132793131, −4.77638244400346045611201975483, −4.02262011158150500625416949266, −2.74707087198032035896045104076, −1.61212059337013755370511986562, −0.58659761012715580114449516196,
0.58659761012715580114449516196, 1.61212059337013755370511986562, 2.74707087198032035896045104076, 4.02262011158150500625416949266, 4.77638244400346045611201975483, 5.82435646862473482612132793131, 6.14458410012205572738532738795, 6.91044421897281929756550200293, 7.62521798233693919309900186987, 8.500444170214735020367757019785