L(s) = 1 | − 5-s − 2.23i·11-s − 4.92i·13-s − 3.83·17-s + 5.96i·19-s − 4.24i·23-s + 25-s + 6.73i·29-s − 1.80i·31-s − 4.32·37-s − 2.50·41-s + 4.96·43-s − 10.8·47-s + 4.50i·53-s + 2.23i·55-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.673i·11-s − 1.36i·13-s − 0.931·17-s + 1.36i·19-s − 0.886i·23-s + 0.200·25-s + 1.25i·29-s − 0.324i·31-s − 0.711·37-s − 0.391·41-s + 0.757·43-s − 1.58·47-s + 0.618i·53-s + 0.301i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8146164486\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8146164486\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 4.92iT - 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 - 5.96iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 - 6.73iT - 29T^{2} \) |
| 31 | \( 1 + 1.80iT - 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 4.50iT - 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 - 6.25iT - 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 8.87iT - 71T^{2} \) |
| 73 | \( 1 + 5.01iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 0.295T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 1.05iT - 97T^{2} \) |
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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
−7.943384777021780845260640505206, −7.34790581996585727777802814810, −6.46930758689394992375844561152, −5.90243536819656259315617900944, −5.15642201065999892382632210754, −4.42320202801249861284536046854, −3.50113667942524204014597495589, −3.05250862881368103417150294832, −1.95521245572495414740297147122, −0.834997819906089008323018067760,
0.22771734098233641415276551736, 1.64658022955209335018379354247, 2.33739575669580314889070589710, 3.32195334965170295803890395109, 4.26572753598472900252478396844, 4.60090027878865132642119534129, 5.42664722889634938758646474398, 6.50611103124636800516356548296, 6.86759608692490998804201193547, 7.46688272037342276690234124681