| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − i·8-s + 9-s − 10-s + 3i·11-s + 12-s + (−3 + 2i)13-s − i·15-s + 16-s − 7·17-s + i·18-s − 3i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.904i·11-s + 0.288·12-s + (−0.832 + 0.554i)13-s − 0.258i·15-s + 0.250·16-s − 1.69·17-s + 0.235i·18-s − 0.688i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2556381021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2556381021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 5 | \( 1 - iT - 5T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 3iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.376975930789174738250102845067, −7.20414748838907514408830424100, −6.91284153487334545829682759258, −6.48874212048975702864790789889, −5.24984449593221129434591017306, −4.79154301357126603320122550906, −4.08478408688125329968607333917, −2.79392930467795604649916398065, −1.78707377797406110995001512741, −0.097638151739995180537399319957,
0.919046021218004238441872688475, 2.12668634279296366397398830420, 3.03859581644279290224801464459, 4.09784159482005571172819105668, 4.75602597361863509500376179575, 5.50715091907045947918358528204, 6.24527560469835591625648185411, 7.11234220890101280371900267171, 8.082222482132109528587908771511, 8.651778179157289819042754179318
