| L(s) = 1 | + 6i·3-s − 706i·7-s + 2.15e3·9-s + 3.84e3·11-s − 4.05e3i·13-s − 858i·17-s + 2.10e4·19-s + 4.23e3·21-s − 8.53e4i·23-s + 2.60e4i·27-s + 8.31e4·29-s + 1.45e5·31-s + 2.30e4i·33-s + 4.98e5i·37-s + 2.43e4·39-s + ⋯ |
| L(s) = 1 | + 0.128i·3-s − 0.777i·7-s + 0.983·9-s + 0.869·11-s − 0.511i·13-s − 0.0423i·17-s + 0.703·19-s + 0.0998·21-s − 1.46i·23-s + 0.254i·27-s + 0.632·29-s + 0.877·31-s + 0.111i·33-s + 1.61i·37-s + 0.0656·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.641689411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641689411\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 706iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 3.84e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.05e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 858iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.53e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 8.31e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.45e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.98e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.67e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.35e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.83e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 6.29e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.37e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.60e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.62e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 4.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.25e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 6.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.97e6iT - 8.07e13T^{2} \) |
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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
−10.18034535860571850733574708646, −9.080196383641798312256534679598, −8.051750225853109995555155764720, −7.05286992875880348838450007107, −6.37040551239256555405289135435, −4.88859636592749308936874488533, −4.13418261999111771847310844318, −3.04245516703898658983458462615, −1.48768689034500290922702207743, −0.62137785189903343839745616042,
1.08319220837784420168341688010, 1.97115994278155239597015655686, 3.35886094047005869693391941490, 4.41375340021742791792672434117, 5.52263687986173720139094077828, 6.57742989355639886412391142932, 7.37357146222017900984629943505, 8.498544558564625215183580834886, 9.432850065831040625720183761916, 10.01684317929532332019209263499
