L(s) = 1 | + (1.79 + 1.79i)3-s + (−3.30 − 3.30i)7-s + 3.44i·9-s + 2.44·11-s + (2.60 + 2.60i)13-s + (−1.48 − 1.48i)17-s + (2.66 − 3.44i)19-s − 11.8i·21-s + (4.78 − 4.78i)23-s + (−0.807 + 0.807i)27-s + 5.32·29-s + 6.52i·31-s + (4.39 + 4.39i)33-s + (7.00 − 7.00i)37-s + 9.34i·39-s + ⋯ |
L(s) = 1 | + (1.03 + 1.03i)3-s + (−1.24 − 1.24i)7-s + 1.14i·9-s + 0.738·11-s + (0.721 + 0.721i)13-s + (−0.359 − 0.359i)17-s + (0.611 − 0.791i)19-s − 2.58i·21-s + (0.997 − 0.997i)23-s + (−0.155 + 0.155i)27-s + 0.989·29-s + 1.17i·31-s + (0.765 + 0.765i)33-s + (1.15 − 1.15i)37-s + 1.49i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.414563824\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.414563824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.66 + 3.44i)T \) |
good | 3 | \( 1 + (-1.79 - 1.79i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.30 + 3.30i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + (-2.60 - 2.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.48 + 1.48i)T + 17iT^{2} \) |
| 23 | \( 1 + (-4.78 + 4.78i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 - 6.52iT - 31T^{2} \) |
| 37 | \( 1 + (-7.00 + 7.00i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.8iT - 41T^{2} \) |
| 43 | \( 1 + (1.81 - 1.81i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.30 - 3.30i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.41 + 3.41i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.52T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + (4.58 - 4.58i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (2.96 - 2.96i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + (-6.93 + 6.93i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.32T + 89T^{2} \) |
| 97 | \( 1 + (-3.77 + 3.77i)T - 97iT^{2} \) |
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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
−9.258353656320053159720325810097, −8.848399902927022545480847593382, −7.78163406527235072806645180445, −6.73469535580949937288272321711, −6.44611116233005578363373823087, −4.71398727234565704763221654446, −4.25041959980227727792505806143, −3.35330377659381249129340114735, −2.81548115313880745201372988361, −0.977808536317234148895461395645,
1.12895041515698391624551416654, 2.30593171978247960467183254878, 3.10269038688149945208662081477, 3.75232838712949233034675058454, 5.42442783710008296405569744865, 6.18420721159369259687066073036, 6.76243500578308019178296130124, 7.72899852041070327816606508831, 8.402863476979913480212426115071, 9.112310904480522339531087101333