L(s) = 1 | − 2.75·3-s − 3.07·5-s + 4.09·7-s + 4.57·9-s + 0.690·11-s − 6.65·13-s + 8.45·15-s − 3.83·17-s − 19-s − 11.2·21-s + 8.12·23-s + 4.44·25-s − 4.33·27-s − 4.25·29-s − 0.724·31-s − 1.89·33-s − 12.5·35-s − 0.241·37-s + 18.3·39-s − 0.711·41-s + 4.10·43-s − 14.0·45-s + 9.79·47-s + 9.76·49-s + 10.5·51-s − 1.44·53-s − 2.12·55-s + ⋯ |
L(s) = 1 | − 1.58·3-s − 1.37·5-s + 1.54·7-s + 1.52·9-s + 0.208·11-s − 1.84·13-s + 2.18·15-s − 0.930·17-s − 0.229·19-s − 2.45·21-s + 1.69·23-s + 0.888·25-s − 0.834·27-s − 0.790·29-s − 0.130·31-s − 0.330·33-s − 2.12·35-s − 0.0396·37-s + 2.93·39-s − 0.111·41-s + 0.625·43-s − 2.09·45-s + 1.42·47-s + 1.39·49-s + 1.47·51-s − 0.199·53-s − 0.285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5228922720\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5228922720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 11 | \( 1 - 0.690T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 23 | \( 1 - 8.12T + 23T^{2} \) |
| 29 | \( 1 + 4.25T + 29T^{2} \) |
| 31 | \( 1 + 0.724T + 31T^{2} \) |
| 37 | \( 1 + 0.241T + 37T^{2} \) |
| 41 | \( 1 + 0.711T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 + 0.864T + 73T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75906550557710901661349512793, −7.32910670521663239074642097832, −6.87864556390784205373464336269, −5.76608312962669532244308662670, −5.01916437039861285888539199949, −4.61253450135818975952314338310, −4.14848300200787022276286537659, −2.73547477185856110821923122617, −1.54964049557611156837623248230, −0.42615859401976145128186180039,
0.42615859401976145128186180039, 1.54964049557611156837623248230, 2.73547477185856110821923122617, 4.14848300200787022276286537659, 4.61253450135818975952314338310, 5.01916437039861285888539199949, 5.76608312962669532244308662670, 6.87864556390784205373464336269, 7.32910670521663239074642097832, 7.75906550557710901661349512793