| L(s) = 1 | + 2.38·5-s + i·7-s + 1.65i·11-s − 0.253i·13-s − 2i·17-s − 7.03·19-s − 2.82·23-s + 0.692·25-s − 1.26·29-s + 0.746i·31-s + 2.38i·35-s − 6.94i·37-s + 5.55i·41-s − 8.67·43-s − 10.9·47-s + ⋯ |
| L(s) = 1 | + 1.06·5-s + 0.377i·7-s + 0.498i·11-s − 0.0702i·13-s − 0.485i·17-s − 1.61·19-s − 0.589·23-s + 0.138·25-s − 0.235·29-s + 0.134i·31-s + 0.403i·35-s − 1.14i·37-s + 0.867i·41-s − 1.32·43-s − 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08267609909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08267609909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.38T + 5T^{2} \) |
| 11 | \( 1 - 1.65iT - 11T^{2} \) |
| 13 | \( 1 + 0.253iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 - 0.746iT - 31T^{2} \) |
| 37 | \( 1 + 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 5.55iT - 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 + 4.53iT - 61T^{2} \) |
| 67 | \( 1 + 0.253T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 - 14.5iT - 79T^{2} \) |
| 83 | \( 1 + 1.77iT - 83T^{2} \) |
| 89 | \( 1 + 3.13iT - 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−7.87792538616704317819726969146, −6.79427932267564328001760423636, −6.39087389482088068413089487006, −5.63181956412380079461314493370, −4.95146253704596022668980671119, −4.17757822463600768920201010357, −3.12697509011909574862885323901, −2.15572453378898266701512831667, −1.70260187352271439341179710699, −0.01796108317131565239751983364,
1.48161806689713880112171711063, 2.12905834446690942954848559879, 3.15564696526782669292391451012, 4.04634647252404850541251064371, 4.80018077174096329067975529866, 5.71122641604759253151212160308, 6.26568557441200888855396945476, 6.76172097782527001723840913326, 7.77829543321615474168486158078, 8.467677329377028578904030794932
