| L(s) = 1 | + (1 − 2i)5-s + i·7-s − 2·11-s + 2i·13-s − 2·19-s − 8i·23-s + (−3 − 4i)25-s + 2·29-s − 6·31-s + (2 + i)35-s − 8i·37-s + 10·41-s − 12i·47-s − 49-s + 2i·53-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.894i)5-s + 0.377i·7-s − 0.603·11-s + 0.554i·13-s − 0.458·19-s − 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s − 1.07·31-s + (0.338 + 0.169i)35-s − 1.31i·37-s + 1.56·41-s − 1.75i·47-s − 0.142·49-s + 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236599872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236599872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.847669678896941153096534877505, −8.046409978528993218381535225911, −7.14742010626432080673601446209, −6.19559177075339810407296268526, −5.55465106819972288041074849825, −4.71590856861349058623844239724, −4.01717151959189970202292248468, −2.61105586429827853467015947303, −1.86623531237938566126083852725, −0.39787492720770672905195314962,
1.40864850050179750847925278859, 2.61202396396576739453234573897, 3.32397422823745551165078787824, 4.33320575000213553308678921560, 5.45528452222121135375912340481, 5.98600831513592375659370259260, 6.96500300420620117042026879621, 7.55639123702979796798360838962, 8.236931855196722988217256065528, 9.377595215854046243735801750644
