| L(s) = 1 | + 0.612·3-s − 5-s + 3.03·7-s − 2.62·9-s − 0.741·11-s + 4.52·13-s − 0.612·15-s + 2.10·17-s − 3.77·19-s + 1.85·21-s + 5.48·23-s + 25-s − 3.44·27-s + 9.90·29-s − 4.71·31-s − 0.454·33-s − 3.03·35-s − 37-s + 2.77·39-s − 2.99·41-s + 12.1·43-s + 2.62·45-s + 5.73·47-s + 2.20·49-s + 1.29·51-s + 9.09·53-s + 0.741·55-s + ⋯ |
| L(s) = 1 | + 0.353·3-s − 0.447·5-s + 1.14·7-s − 0.874·9-s − 0.223·11-s + 1.25·13-s − 0.158·15-s + 0.511·17-s − 0.866·19-s + 0.405·21-s + 1.14·23-s + 0.200·25-s − 0.663·27-s + 1.83·29-s − 0.846·31-s − 0.0790·33-s − 0.512·35-s − 0.164·37-s + 0.444·39-s − 0.468·41-s + 1.85·43-s + 0.391·45-s + 0.836·47-s + 0.314·49-s + 0.180·51-s + 1.24·53-s + 0.0999·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.985013247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985013247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 0.612T + 3T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 41 | \( 1 + 2.99T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 - 9.09T + 53T^{2} \) |
| 59 | \( 1 - 5.06T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 8.15T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−9.176174203422804490333033838788, −8.557098295176781875045190882688, −8.128407746122018263486218149242, −7.25386664452878993629771786091, −6.16880926361256912263456970868, −5.32110842242707418188179985644, −4.40346158587637574309514131906, −3.43915397723303425855806444210, −2.41340336392409779298030448841, −1.04745989450988111344053866161,
1.04745989450988111344053866161, 2.41340336392409779298030448841, 3.43915397723303425855806444210, 4.40346158587637574309514131906, 5.32110842242707418188179985644, 6.16880926361256912263456970868, 7.25386664452878993629771786091, 8.128407746122018263486218149242, 8.557098295176781875045190882688, 9.176174203422804490333033838788
