L(s) = 1 | + 2.66i·5-s + i·7-s − 0.164·11-s + 3.56·13-s + 3.41i·17-s − 3.89i·19-s − 5.29·23-s − 2.09·25-s − 7.84i·29-s + 8.72i·31-s − 2.66·35-s + 7.82·37-s + 1.62i·41-s + 11.2i·43-s + 0.585·47-s + ⋯ |
L(s) = 1 | + 1.19i·5-s + 0.377i·7-s − 0.0496·11-s + 0.987·13-s + 0.828i·17-s − 0.894i·19-s − 1.10·23-s − 0.419·25-s − 1.45i·29-s + 1.56i·31-s − 0.450·35-s + 1.28·37-s + 0.254i·41-s + 1.71i·43-s + 0.0854·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.583450791\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.583450791\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.66iT - 5T^{2} \) |
| 11 | \( 1 + 0.164T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 7.84iT - 29T^{2} \) |
| 31 | \( 1 - 8.72iT - 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 1.62iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 0.585T + 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 6.83iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 + 9.09iT - 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 + 6.05iT - 89T^{2} \) |
| 97 | \( 1 - 4.72T + 97T^{2} \) |
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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
−8.161856952493609494258654035367, −7.75354463097031516357968559692, −6.73281517409710517799392586214, −6.27285826258043143981171134525, −5.76761488961815591007168285548, −4.60756035188174838147868301073, −3.88666912772314143536615065701, −3.00087245902669297926527335831, −2.41009083856948803308834011849, −1.25279091536722655624917797072,
0.42933589368998279855420924295, 1.33685097044150679318386684295, 2.28316888101350791962345837764, 3.62212109421637180606533628657, 4.04590181763966271908296242441, 5.00390960398924798998189733355, 5.55419762739085094878910836781, 6.32930073923248399633413237953, 7.14316149385782528710324611892, 8.105315570513899256424551781891