L(s) = 1 | + (−1.41 − 1.73i)5-s − 2.82i·7-s + 2i·11-s − 2.82·13-s + 4.89i·17-s + 5.65i·23-s + (−0.999 + 4.89i)25-s − 3.46i·29-s − 3.46·31-s + (−4.89 + 4.00i)35-s − 2.82·37-s + 8·41-s − 9.79·43-s − 1.00·49-s + 8.48·53-s + ⋯ |
L(s) = 1 | + (−0.632 − 0.774i)5-s − 1.06i·7-s + 0.603i·11-s − 0.784·13-s + 1.18i·17-s + 1.17i·23-s + (−0.199 + 0.979i)25-s − 0.643i·29-s − 0.622·31-s + (−0.828 + 0.676i)35-s − 0.464·37-s + 1.24·41-s − 1.49·43-s − 0.142·49-s + 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9711233414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9711233414\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 + 1.73i)T \) |
good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.866346682214797650261317298621, −7.961146619252180858966514005803, −7.50133772685502574412302819437, −6.83700520794359891558622149266, −5.69798696299943447877295061134, −4.89892124339068306994349358758, −4.08809189319539835345140648726, −3.59547353107644370355187179931, −2.07766425498228054576991587953, −0.999165412196470806890231030723,
0.36321492407514021181108398389, 2.25319450942734545035795054039, 2.85908535234159500282806897032, 3.74754675176867385883179853437, 4.88613817155461378267590181492, 5.50388382486387952411279103937, 6.56130017232237653968403762376, 7.03987328259177588915306084382, 7.975866844803467001118572242149, 8.584408821685585660379240281773