L(s) = 1 | + (1.73 − 1.41i)5-s + 4.24·7-s + 2.44·11-s + 2.44i·13-s + 6.92·17-s − 6.92i·19-s + 6i·23-s + (0.999 − 4.89i)25-s + 2.82i·29-s + 3.46i·31-s + (7.34 − 6i)35-s − 2.44i·37-s + 7.07i·41-s − 8.48·43-s + 10.9·49-s + ⋯ |
L(s) = 1 | + (0.774 − 0.632i)5-s + 1.60·7-s + 0.738·11-s + 0.679i·13-s + 1.68·17-s − 1.58i·19-s + 1.25i·23-s + (0.199 − 0.979i)25-s + 0.525i·29-s + 0.622i·31-s + (1.24 − 1.01i)35-s − 0.402i·37-s + 1.10i·41-s − 1.29·43-s + 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.955404661\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.955404661\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 + 1.41i)T \) |
good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2.44iT - 37T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−8.818084234347214342998341847657, −8.052563312892840757565486752948, −7.31343323685027353714522250179, −6.44261949491222324514489588740, −5.39837317906916589394035478346, −5.01031774097288204159405101615, −4.20873159329675997782244877521, −2.99339865455116542788334488847, −1.64718594128255049594050552311, −1.26741593950734141883962317805,
1.23729459403020196778518069708, 1.96096036087451722990762698319, 3.11829462171938108531380550501, 4.04617783097859621490653280037, 5.07828041404656393087502751878, 5.73941875804490152667026417120, 6.37670908895291309766044712418, 7.51581997259581767807100445172, 7.957628890531697224471475718850, 8.678966369227428959758158101374