L(s) = 1 | + 2-s + 3-s + 4-s + 0.927·5-s + 6-s + 1.16·7-s + 8-s + 9-s + 0.927·10-s + 3.50·11-s + 12-s + 0.153·13-s + 1.16·14-s + 0.927·15-s + 16-s + 4.78·17-s + 18-s − 19-s + 0.927·20-s + 1.16·21-s + 3.50·22-s + 9.11·23-s + 24-s − 4.13·25-s + 0.153·26-s + 27-s + 1.16·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.414·5-s + 0.408·6-s + 0.438·7-s + 0.353·8-s + 0.333·9-s + 0.293·10-s + 1.05·11-s + 0.288·12-s + 0.0425·13-s + 0.310·14-s + 0.239·15-s + 0.250·16-s + 1.16·17-s + 0.235·18-s − 0.229·19-s + 0.207·20-s + 0.253·21-s + 0.747·22-s + 1.90·23-s + 0.204·24-s − 0.827·25-s + 0.0300·26-s + 0.192·27-s + 0.219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.378113082\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.378113082\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 0.927T + 5T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.153T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 23 | \( 1 - 9.11T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 3.95T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 0.967T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.910413771195292667141366818678, −7.38829447622468639924090878957, −6.58592862858968966692363062330, −5.90092404570064506957184593648, −5.15226126076862051968265710419, −4.40469654902127630110533225147, −3.58899756678130935510859600551, −2.97274024426105687355593177745, −1.88703687734192841277064287689, −1.20161474345917723160803058552,
1.20161474345917723160803058552, 1.88703687734192841277064287689, 2.97274024426105687355593177745, 3.58899756678130935510859600551, 4.40469654902127630110533225147, 5.15226126076862051968265710419, 5.90092404570064506957184593648, 6.58592862858968966692363062330, 7.38829447622468639924090878957, 7.910413771195292667141366818678