| L(s) = 1 | + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s − 2i·7-s − i·8-s − 9-s + (1 + 2i)10-s − i·12-s − 2i·13-s + 2·14-s + (1 + 2i)15-s + 16-s − i·18-s + 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s − 0.288i·12-s − 0.554i·13-s + 0.534·14-s + (0.258 + 0.516i)15-s + 0.250·16-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70112 + 0.401581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70112 + 0.401581i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−10.19162207819299726554368610715, −9.237516804256183078672722751031, −8.537812777193137333475258810287, −7.60075428088826133685036102694, −6.63412707880630250998924878427, −5.72811720796610628786290294047, −4.97314598266374029261205774366, −4.11181043515319211873691267605, −2.81595747864006454330835131972, −0.967578865639452462690145627241,
1.36574474267146578414953301696, 2.41471440131578098084008800367, 3.21238279759276996273909879409, 4.75980100681370702151201742715, 5.70918964423839515412498593212, 6.47160133621760690562630147376, 7.45917870493747497762657713033, 8.569801162487930724931030941037, 9.274767585490906418972086826201, 10.00047568405200539228697810234
