L(s) = 1 | + (0.928 − 0.928i)2-s + (−1.37 − 1.05i)3-s + 0.276i·4-s + (2.12 − 2.12i)5-s + (−2.25 + 0.293i)6-s + (2.06 − 2.06i)7-s + (2.11 + 2.11i)8-s + (0.767 + 2.90i)9-s − 3.94i·10-s + (1.88 + 1.88i)11-s + (0.291 − 0.379i)12-s − 3.83i·14-s + (−5.16 + 0.671i)15-s + 3.37·16-s + 0.198·17-s + (3.40 + 1.98i)18-s + ⋯ |
L(s) = 1 | + (0.656 − 0.656i)2-s + (−0.792 − 0.610i)3-s + 0.138i·4-s + (0.950 − 0.950i)5-s + (−0.920 + 0.119i)6-s + (0.780 − 0.780i)7-s + (0.747 + 0.747i)8-s + (0.255 + 0.966i)9-s − 1.24i·10-s + (0.568 + 0.568i)11-s + (0.0842 − 0.109i)12-s − 1.02i·14-s + (−1.33 + 0.173i)15-s + 0.842·16-s + 0.0482·17-s + (0.802 + 0.466i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43186 - 1.44994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43186 - 1.44994i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.37 + 1.05i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.928 + 0.928i)T - 2iT^{2} \) |
| 5 | \( 1 + (-2.12 + 2.12i)T - 5iT^{2} \) |
| 7 | \( 1 + (-2.06 + 2.06i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.88 - 1.88i)T + 11iT^{2} \) |
| 17 | \( 1 - 0.198T + 17T^{2} \) |
| 19 | \( 1 + (3.75 + 3.75i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 + 3.73iT - 29T^{2} \) |
| 31 | \( 1 + (-0.550 - 0.550i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.60 - 3.60i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.69 - 2.69i)T - 41iT^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 + (-4.29 - 4.29i)T + 47iT^{2} \) |
| 53 | \( 1 - 13.6iT - 53T^{2} \) |
| 59 | \( 1 + (2.36 + 2.36i)T + 59iT^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + (-5.33 - 5.33i)T + 67iT^{2} \) |
| 71 | \( 1 + (3.78 - 3.78i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.46 + 3.46i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + (1.84 - 1.84i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.776 + 0.776i)T + 89iT^{2} \) |
| 97 | \( 1 + (-8.60 - 8.60i)T + 97iT^{2} \) |
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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
−10.88261854811920375819331984979, −10.18018039723633150654700129585, −8.885690847205571661293027743411, −7.902231804409031009363809678894, −6.96220784112052689657845979392, −5.77319312371933257610586110926, −4.78052172749322086259042023364, −4.25730382377750002943264211063, −2.20940547336661745047875790972, −1.32767855862334024971008426756,
1.81407869116434658819789541082, 3.62109483445524268991347905491, 4.80536607738604064787716223104, 5.76202590096403062691811913157, 6.13399640481942184408801124178, 6.98088903781774988601365790201, 8.470319233434678909781224306203, 9.619545733537231157799732012086, 10.34662819234942939477266290546, 11.00036140137645342390140100966