L(s) = 1 | − 0.618·2-s − 1.61·4-s + 5-s − 3.23·7-s + 2.23·8-s − 0.618·10-s − 5.23·13-s + 2.00·14-s + 1.85·16-s − 5.47·17-s − 6.47·19-s − 1.61·20-s + 4.70·23-s + 25-s + 3.23·26-s + 5.23·28-s − 1.23·29-s − 6.70·31-s − 5.61·32-s + 3.38·34-s − 3.23·35-s + 0.763·37-s + 4.00·38-s + 2.23·40-s + 3.52·41-s + 5.23·43-s − 2.90·46-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.447·5-s − 1.22·7-s + 0.790·8-s − 0.195·10-s − 1.45·13-s + 0.534·14-s + 0.463·16-s − 1.32·17-s − 1.48·19-s − 0.361·20-s + 0.981·23-s + 0.200·25-s + 0.634·26-s + 0.989·28-s − 0.229·29-s − 1.20·31-s − 0.993·32-s + 0.580·34-s − 0.546·35-s + 0.125·37-s + 0.648·38-s + 0.353·40-s + 0.550·41-s + 0.798·43-s − 0.429·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3719663722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3719663722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.324387151799127760147034121650, −7.43896861292988426411055241229, −6.75654219947332354995562589568, −6.17076113731847744224041122083, −5.13559460185910387870083039428, −4.58430576136809789874024714360, −3.73522973404937058473185919824, −2.72061068079549631747266905345, −1.88819327925384060815489269683, −0.33425767859809056623056024827,
0.33425767859809056623056024827, 1.88819327925384060815489269683, 2.72061068079549631747266905345, 3.73522973404937058473185919824, 4.58430576136809789874024714360, 5.13559460185910387870083039428, 6.17076113731847744224041122083, 6.75654219947332354995562589568, 7.43896861292988426411055241229, 8.324387151799127760147034121650