L(s) = 1 | + 7.34i·5-s − 10.3·7-s + 8.48i·11-s − 10.3·13-s − 21.2i·17-s + 20·19-s − 14.6i·23-s − 29·25-s + 36.7i·29-s − 51.9·31-s − 76.3i·35-s − 41.5·37-s + 72.1i·41-s − 40·43-s − 73.4i·47-s + ⋯ |
L(s) = 1 | + 1.46i·5-s − 1.48·7-s + 0.771i·11-s − 0.799·13-s − 1.24i·17-s + 1.05·19-s − 0.638i·23-s − 1.15·25-s + 1.26i·29-s − 1.67·31-s − 2.18i·35-s − 1.12·37-s + 1.75i·41-s − 0.930·43-s − 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5286133472\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5286133472\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 7.34iT - 25T^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10.3T + 169T^{2} \) |
| 17 | \( 1 + 21.2iT - 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 - 36.7iT - 841T^{2} \) |
| 31 | \( 1 + 51.9T + 961T^{2} \) |
| 37 | \( 1 + 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 36.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 100T + 4.48e3T^{2} \) |
| 71 | \( 1 + 73.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 40T + 9.40e3T^{2} \) |
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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
−8.938406896565293672913926377235, −7.54594962939954147961342826748, −6.94846568788374562564860625471, −6.79671726993344826352599865943, −5.64102413480843861107871154739, −4.76092706813331023483617193627, −3.28921886309282850884478807122, −3.15966650804142956670537742611, −2.06327976465073044908704163989, −0.17003908418920058019698331097,
0.77371257571262982094640833858, 2.02139314297924976916718400226, 3.40076810202251057683538971862, 3.91644348636113494274098393692, 5.20310637258927422168880035640, 5.63946516508040662483600162842, 6.53348802517850821345811399324, 7.48181341064953476015655335579, 8.287749212219646088173110683117, 9.058438279505439907639347252201