| L(s) = 1 | − i·2-s − i·3-s + 4-s + (−2 − i)5-s − 6-s − i·7-s − 3i·8-s − 9-s + (−1 + 2i)10-s − 11-s − i·12-s − 4i·13-s − 14-s + (−1 + 2i)15-s − 16-s − 2i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s + (−0.894 − 0.447i)5-s − 0.408·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s − 0.301·11-s − 0.288i·12-s − 1.10i·13-s − 0.267·14-s + (−0.258 + 0.516i)15-s − 0.250·16-s − 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125301165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125301165\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.462436573210190299505018527905, −8.359359965440661790884563067014, −7.55763752519512470532902732689, −7.15352748444524550193457558470, −5.95001249438461852111411658082, −4.94742237210942171762763709176, −3.66174311639597372752432973761, −3.01302609659908305534976401763, −1.64238382775664121014868067820, −0.46675791395263709011239999340,
2.13665514533546968980186342720, 3.21247672631017638303446338122, 4.30635701739772528425980030419, 5.17844986001079886714265213165, 6.32824478092208750211776802413, 6.80940568144947170766623991882, 7.83466421962417988681777630049, 8.428458362809223827938254670442, 9.261437518852549106443109776454, 10.51113421828337462823454098155
