| L(s) = 1 | + 2i·5-s − 8i·7-s − 4·11-s − 14i·13-s − 18·17-s − 12·19-s − 40i·23-s + 21·25-s + 14i·29-s + 32i·31-s + 16·35-s + 30i·37-s − 14·41-s − 28·43-s + 16i·47-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s − 1.14i·7-s − 0.363·11-s − 1.07i·13-s − 1.05·17-s − 0.631·19-s − 1.73i·23-s + 0.839·25-s + 0.482i·29-s + 1.03i·31-s + 0.457·35-s + 0.810i·37-s − 0.341·41-s − 0.651·43-s + 0.340i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1832896863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1832896863\) |
| \(L(2)\) |
|
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 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 + 8iT - 49T^{2} \) |
| 11 | \( 1 + 4T + 121T^{2} \) |
| 13 | \( 1 + 14iT - 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 + 40iT - 529T^{2} \) |
| 29 | \( 1 - 14iT - 841T^{2} \) |
| 31 | \( 1 - 32iT - 961T^{2} \) |
| 37 | \( 1 - 30iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 52T + 3.48e3T^{2} \) |
| 61 | \( 1 - 82iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 56iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 66T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 140T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30T + 7.92e3T^{2} \) |
| 97 | \( 1 + 14T + 9.40e3T^{2} \) |
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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
−8.942707034355092813912025595740, −8.370181052562010291334944506929, −7.50258737066956086574066977931, −6.79241488472659275772393882685, −6.22469679367292626025326881674, −4.95293463466037859879983141279, −4.39759605720404394077324318651, −3.29073264636413013735897090367, −2.52050534777745948057186738638, −1.08540858575378585907362876523,
0.04638254913553108880378809455, 1.75178114036350055355328843710, 2.41655184693498554699568449575, 3.65673098220255766062119950968, 4.61554565046261241449438482568, 5.34848685107439339589964499238, 6.16535887189092893500864498551, 6.91496771746120635167912446577, 7.889467957867584528902136002756, 8.697731638327443359063671448394
