| L(s) = 1 | − 1.40i·2-s + i·3-s + 0.0352·4-s + (−2.22 + 0.223i)5-s + 1.40·6-s + i·7-s − 2.85i·8-s − 9-s + (0.313 + 3.11i)10-s + 11-s + 0.0352i·12-s + 1.11i·13-s + 1.40·14-s + (−0.223 − 2.22i)15-s − 3.92·16-s + 3.90i·17-s + ⋯ |
| L(s) = 1 | − 0.991i·2-s + 0.577i·3-s + 0.0176·4-s + (−0.994 + 0.0999i)5-s + 0.572·6-s + 0.377i·7-s − 1.00i·8-s − 0.333·9-s + (0.0990 + 0.986i)10-s + 0.301·11-s + 0.0101i·12-s + 0.309i·13-s + 0.374·14-s + (−0.0577 − 0.574i)15-s − 0.982·16-s + 0.946i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.0999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.0999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.447569427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447569427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.22 - 0.223i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.40iT - 2T^{2} \) |
| 13 | \( 1 - 1.11iT - 13T^{2} \) |
| 17 | \( 1 - 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 6.16T + 19T^{2} \) |
| 23 | \( 1 - 5.20iT - 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 8.95T + 31T^{2} \) |
| 37 | \( 1 - 0.540iT - 37T^{2} \) |
| 41 | \( 1 - 3.36T + 41T^{2} \) |
| 43 | \( 1 - 9.48iT - 43T^{2} \) |
| 47 | \( 1 - 3.04iT - 47T^{2} \) |
| 53 | \( 1 - 2.86iT - 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 5.35iT - 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 0.333iT - 73T^{2} \) |
| 79 | \( 1 + 0.575T + 79T^{2} \) |
| 83 | \( 1 + 0.318iT - 83T^{2} \) |
| 89 | \( 1 + 6.02T + 89T^{2} \) |
| 97 | \( 1 - 0.903iT - 97T^{2} \) |
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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
−9.816337612584727577427088467050, −9.358709111096890780046532269190, −8.256229216388457669702206942660, −7.44681362247020152893877108755, −6.47798867003634695173035255711, −5.37763870102371950440825677889, −4.16000058767211368220102686992, −3.58706018548204600523219398647, −2.67884720784436040688637248220, −1.23149916713470849924130995468,
0.72361933814224443111248512266, 2.48203029886106150636261167863, 3.64628878634945629892422987921, 4.87730469575386157265989544690, 5.65020972962621988067955986874, 6.85968295004241891963392170945, 7.14105125955825810611334858085, 7.958370904177123204259329656035, 8.521571483557856834256010515415, 9.567647097310028361360957787973
