L(s) = 1 | + 2.30i·2-s + i·3-s − 3.33·4-s + 3.77i·5-s − 2.30·6-s + (−0.474 − 2.60i)7-s − 3.07i·8-s − 9-s − 8.71·10-s + (3.23 − 0.726i)11-s − 3.33i·12-s + 3.32·13-s + (6.01 − 1.09i)14-s − 3.77·15-s + 0.441·16-s − 5.12·17-s + ⋯ |
L(s) = 1 | + 1.63i·2-s + 0.577i·3-s − 1.66·4-s + 1.68i·5-s − 0.942·6-s + (−0.179 − 0.983i)7-s − 1.08i·8-s − 0.333·9-s − 2.75·10-s + (0.975 − 0.219i)11-s − 0.962i·12-s + 0.923·13-s + (1.60 − 0.293i)14-s − 0.974·15-s + 0.110·16-s − 1.24·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.221459 - 1.08878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.221459 - 1.08878i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.474 + 2.60i)T \) |
| 11 | \( 1 + (-3.23 + 0.726i)T \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 5 | \( 1 - 3.77iT - 5T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 4.27iT - 29T^{2} \) |
| 31 | \( 1 - 1.11iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 - 6.74iT - 43T^{2} \) |
| 47 | \( 1 + 5.58iT - 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 7.36iT - 59T^{2} \) |
| 61 | \( 1 + 9.23T + 61T^{2} \) |
| 67 | \( 1 - 1.96T + 67T^{2} \) |
| 71 | \( 1 + 5.07T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 5.88iT - 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + 14.4iT - 97T^{2} \) |
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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
−13.46878057296973023806226225409, −11.35199072052479354129575235109, −10.81118927787820004888104488142, −9.664310508750690813252023713888, −8.644169469114574856479396235774, −7.37883275700495115624260127878, −6.70769146018569985741539596592, −6.05128643482724763708010680238, −4.41007214022639243032774049690, −3.35683391053267246093987693102,
1.02087509614568616118788150341, 2.19196838071944143170220499186, 3.88161114935368155822888232514, 4.99704511659237160517070439240, 6.31681158148608567187328783703, 8.268275426130004024181608216589, 9.103022560231556958525979103199, 9.442383449425472096509314203243, 11.11559384815169385944848139976, 11.85886592168349038077974400262