L(s) = 1 | − 1.12·2-s − 3-s − 0.744·4-s + (−1.27 − 1.83i)5-s + 1.12·6-s + (2.54 + 0.709i)7-s + 3.07·8-s + 9-s + (1.42 + 2.05i)10-s + (2.57 − 2.09i)11-s + 0.744·12-s − 0.476i·13-s + (−2.85 − 0.794i)14-s + (1.27 + 1.83i)15-s − 1.95·16-s + 4.13i·17-s + ⋯ |
L(s) = 1 | − 0.792·2-s − 0.577·3-s − 0.372·4-s + (−0.569 − 0.822i)5-s + 0.457·6-s + (0.963 + 0.268i)7-s + 1.08·8-s + 0.333·9-s + (0.451 + 0.651i)10-s + (0.775 − 0.631i)11-s + 0.214·12-s − 0.132i·13-s + (−0.763 − 0.212i)14-s + (0.328 + 0.474i)15-s − 0.489·16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7478881691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7478881691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (1.27 + 1.83i)T \) |
| 7 | \( 1 + (-2.54 - 0.709i)T \) |
| 11 | \( 1 + (-2.57 + 2.09i)T \) |
good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 13 | \( 1 + 0.476iT - 13T^{2} \) |
| 17 | \( 1 - 4.13iT - 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 - 2.30iT - 23T^{2} \) |
| 29 | \( 1 + 8.78iT - 29T^{2} \) |
| 31 | \( 1 - 7.12iT - 31T^{2} \) |
| 37 | \( 1 - 4.59iT - 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 4.46iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.29iT - 73T^{2} \) |
| 79 | \( 1 - 0.726iT - 79T^{2} \) |
| 83 | \( 1 + 8.78iT - 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−9.487745373546458836698366902848, −8.865120244475287773657673426313, −8.157812977642148708525773388414, −7.69176028343666462322710487311, −6.34294513732868805641359563205, −5.39859753107119603953123025902, −4.49988123486603795114041458148, −3.88832097760844767446594535750, −1.73200172742888386863893781319, −0.74713588724047149833131523745,
0.847453050293357844652102070728, 2.25161540789522114698179660235, 4.06171356559862606134157110929, 4.46223865427130998772719516888, 5.63209571707939435653714158813, 7.02163730396470362111365184186, 7.26839801297976134459992469580, 8.241264619390040630155626999016, 9.076460123390178802921532646948, 9.872765897142391810138238297564