L(s) = 1 | − 3-s + 2i·5-s + 2i·7-s + 9-s + (3 − 2i)13-s − 2i·15-s − 2·17-s − 6i·19-s − 2i·21-s + 4·23-s + 25-s − 27-s + 10·29-s − 10i·31-s − 4·35-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894i·5-s + 0.755i·7-s + 0.333·9-s + (0.832 − 0.554i)13-s − 0.516i·15-s − 0.485·17-s − 1.37i·19-s − 0.436i·21-s + 0.834·23-s + 0.200·25-s − 0.192·27-s + 1.85·29-s − 1.79i·31-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.551088412\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.551088412\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.945788036158832306267704992298, −8.314867923470759919370174204799, −7.25906597754550183834240310899, −6.59696483526209601878506200631, −6.02439281888965314938366131321, −5.13575354460651775619037103489, −4.30043225343445173569649623752, −3.04904025041904822423244154549, −2.46411762025911602682651204632, −0.862667892041580733144885697872,
0.843748852506503738779317108194, 1.63894762662295027536884512983, 3.27904487397404063696102251563, 4.22500968495076618686509278589, 4.83787937079836053884547096337, 5.67672145735438302741027594839, 6.62293506178312334438084101834, 7.10921587357757010295235799710, 8.367517411383458877498500146946, 8.622705684907070998753851757528