| L(s) = 1 | − 3·3-s − 11.6·7-s + 9·9-s + 41.5·11-s − 1.76·13-s + 63.3·17-s + 25.3·19-s + 34.9·21-s + 181.·23-s − 27·27-s + 204.·29-s − 61.8·31-s − 124.·33-s − 365.·37-s + 5.28·39-s + 130.·41-s + 135.·43-s − 380.·47-s − 206.·49-s − 190.·51-s − 352.·53-s − 76.0·57-s + 556.·59-s + 88.5·61-s − 104.·63-s − 708.·67-s − 545.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.629·7-s + 0.333·9-s + 1.13·11-s − 0.0375·13-s + 0.904·17-s + 0.306·19-s + 0.363·21-s + 1.64·23-s − 0.192·27-s + 1.31·29-s − 0.358·31-s − 0.657·33-s − 1.62·37-s + 0.0217·39-s + 0.497·41-s + 0.479·43-s − 1.18·47-s − 0.603·49-s − 0.522·51-s − 0.914·53-s − 0.176·57-s + 1.22·59-s + 0.185·61-s − 0.209·63-s − 1.29·67-s − 0.952·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.901735875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.901735875\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 11.6T + 343T^{2} \) |
| 11 | \( 1 - 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.76T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 352.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 556.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 88.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 708.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 750.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 57.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 544.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 533.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 300.T + 9.12e5T^{2} \) |
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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
−8.763176372006467409407517998284, −7.74031053831974532079479294328, −6.81096410520763443047786235479, −6.47813846877170481053037351380, −5.45588277566059907832230055644, −4.76540563292474538222437505717, −3.67773955415033496183722574360, −2.99773736677300804726985090462, −1.52075946442055809857801053909, −0.67812223412743517116806039187,
0.67812223412743517116806039187, 1.52075946442055809857801053909, 2.99773736677300804726985090462, 3.67773955415033496183722574360, 4.76540563292474538222437505717, 5.45588277566059907832230055644, 6.47813846877170481053037351380, 6.81096410520763443047786235479, 7.74031053831974532079479294328, 8.763176372006467409407517998284
