| L(s) = 1 | + (−1 − 2i)5-s + (3 − 3i)7-s − 2i·11-s + (3 − 3i)13-s + (−1 − i)17-s − 4·19-s + (−1 − i)23-s + (−3 + 4i)25-s + 10i·31-s + (−9 − 3i)35-s + (−1 − i)37-s + 10·41-s + (−5 − 5i)43-s + (−3 + 3i)47-s − 11i·49-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + (1.13 − 1.13i)7-s − 0.603i·11-s + (0.832 − 0.832i)13-s + (−0.242 − 0.242i)17-s − 0.917·19-s + (−0.208 − 0.208i)23-s + (−0.600 + 0.800i)25-s + 1.79i·31-s + (−1.52 − 0.507i)35-s + (−0.164 − 0.164i)37-s + 1.56·41-s + (−0.762 − 0.762i)43-s + (−0.437 + 0.437i)47-s − 1.57i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520912195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520912195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + (5 + 5i)T + 43iT^{2} \) |
| 47 | \( 1 + (3 - 3i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 - 5i)T + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−8.964785749277635890832546438725, −8.393333027030625546316945048720, −7.86206541664079942653056943971, −6.96761527358850512889640921527, −5.83676401749638405650404396099, −4.90981333811031777935908377271, −4.23910319679701271269130186929, −3.34558653352217472077771431651, −1.62055335109325275406319653114, −0.63025091104298254046510270754,
1.80990077382542413335340514302, 2.54751085984688238313808261551, 3.95199397086903270015799150250, 4.61103831015660639092300051832, 5.85699954450988039942966701777, 6.44017783180783440629036759724, 7.50652372250967064258577505176, 8.177414983583069768730142091613, 8.896411573292200335676499962939, 9.748825770483848995467634465464
