L(s) = 1 | + 2-s + 1.61·3-s + 4-s + 5-s + 1.61·6-s − 4.97·7-s + 8-s − 0.390·9-s + 10-s + 2.05·11-s + 1.61·12-s + 6.01·13-s − 4.97·14-s + 1.61·15-s + 16-s − 6.28·17-s − 0.390·18-s + 6.01·19-s + 20-s − 8.02·21-s + 2.05·22-s + 2.50·23-s + 1.61·24-s + 25-s + 6.01·26-s − 5.47·27-s − 4.97·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.932·3-s + 0.5·4-s + 0.447·5-s + 0.659·6-s − 1.87·7-s + 0.353·8-s − 0.130·9-s + 0.316·10-s + 0.620·11-s + 0.466·12-s + 1.66·13-s − 1.32·14-s + 0.417·15-s + 0.250·16-s − 1.52·17-s − 0.0921·18-s + 1.37·19-s + 0.223·20-s − 1.75·21-s + 0.438·22-s + 0.521·23-s + 0.329·24-s + 0.200·25-s + 1.18·26-s − 1.05·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.275400901\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.275400901\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 - 6.01T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 - 0.373T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 5.27T + 41T^{2} \) |
| 43 | \( 1 - 0.00845T + 43T^{2} \) |
| 47 | \( 1 + 0.449T + 47T^{2} \) |
| 53 | \( 1 - 1.61T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 - 8.17T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.148584333701613424031459433639, −7.14214871843457477575116522439, −6.44053295714990898756830492039, −6.17722075848719920964957871999, −5.31076258129172258528887781867, −4.07513711228052572761186622164, −3.55571536007184659741970029369, −2.97106466509277232248158780866, −2.25678391192397271625496860139, −0.941667506874337222414320434297,
0.941667506874337222414320434297, 2.25678391192397271625496860139, 2.97106466509277232248158780866, 3.55571536007184659741970029369, 4.07513711228052572761186622164, 5.31076258129172258528887781867, 6.17722075848719920964957871999, 6.44053295714990898756830492039, 7.14214871843457477575116522439, 8.148584333701613424031459433639