L(s) = 1 | + 2-s + 3-s + 4-s + 0.771·5-s + 6-s − 0.402·7-s + 8-s + 9-s + 0.771·10-s + 3.35·11-s + 12-s − 13-s − 0.402·14-s + 0.771·15-s + 16-s + 2.85·17-s + 18-s − 3.72·19-s + 0.771·20-s − 0.402·21-s + 3.35·22-s + 5.56·23-s + 24-s − 4.40·25-s − 26-s + 27-s − 0.402·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.344·5-s + 0.408·6-s − 0.152·7-s + 0.353·8-s + 0.333·9-s + 0.243·10-s + 1.01·11-s + 0.288·12-s − 0.277·13-s − 0.107·14-s + 0.199·15-s + 0.250·16-s + 0.692·17-s + 0.235·18-s − 0.854·19-s + 0.172·20-s − 0.0878·21-s + 0.715·22-s + 1.16·23-s + 0.204·24-s − 0.880·25-s − 0.196·26-s + 0.192·27-s − 0.0760·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.973889965\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.973889965\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 0.771T + 5T^{2} \) |
| 7 | \( 1 + 0.402T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 - 2.39T + 53T^{2} \) |
| 59 | \( 1 + 0.522T + 59T^{2} \) |
| 61 | \( 1 + 0.0978T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 + 7.64T + 97T^{2} \) |
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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
−7.72889756563402172832405150370, −7.07940403058559963816275839543, −6.33039369881984247809004593669, −5.87590302937743639794020502643, −4.87456963772603739875740766952, −4.25792904278386950030516193376, −3.52731428729945633050245222009, −2.77511455943902069011186485211, −1.97559725900928472278163484649, −1.01204457945034578346561775095,
1.01204457945034578346561775095, 1.97559725900928472278163484649, 2.77511455943902069011186485211, 3.52731428729945633050245222009, 4.25792904278386950030516193376, 4.87456963772603739875740766952, 5.87590302937743639794020502643, 6.33039369881984247809004593669, 7.07940403058559963816275839543, 7.72889756563402172832405150370