| L(s) = 1 | + 10.3i·5-s − 3.46·7-s − 55.4i·13-s + 90·17-s + 116i·19-s + 103.·23-s + 17·25-s + 259. i·29-s − 301.·31-s − 36i·35-s − 34.6i·37-s + 54·41-s − 20i·43-s − 394.·47-s − 331·49-s + ⋯ |
| L(s) = 1 | + 0.929i·5-s − 0.187·7-s − 1.18i·13-s + 1.28·17-s + 1.40i·19-s + 0.942·23-s + 0.136·25-s + 1.66i·29-s − 1.74·31-s − 0.173i·35-s − 0.153i·37-s + 0.205·41-s − 0.0709i·43-s − 1.22·47-s − 0.965·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.529147235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529147235\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 10.3iT - 125T^{2} \) |
| 7 | \( 1 + 3.46T + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 55.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 90T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.6iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 54T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 324iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 575. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 116iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 148.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190T + 9.12e5T^{2} \) |
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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
−10.53903015812618728646390231150, −9.930498416260379799636453383480, −8.811742263318790341614417005632, −7.73566486170158603610320871568, −7.12042485671404348946149375329, −5.95997791176097202630506000045, −5.21441709402519607279620709709, −3.54770112797699938034778810813, −2.99269839248672319317280663115, −1.32586054326697453404228349761,
0.48208755968438760267178356362, 1.79587838159385328714875736922, 3.28854477800799987510054958763, 4.53003537850584610330309770400, 5.26304131498748058550176080354, 6.45509157921449141693658915182, 7.38065961444394514039507099289, 8.422986098828678696064980109813, 9.258989664993869457861840628430, 9.777744604769517788075418966219
