| L(s) = 1 | + 2-s + 4-s + 2.23i·5-s + (1.58 − 2.12i)7-s + 8-s + 2.23i·10-s − 1.41i·11-s + 3.16·13-s + (1.58 − 2.12i)14-s + 16-s + 4.47i·17-s + 2.23i·20-s − 1.41i·22-s + 6·23-s − 5.00·25-s + 3.16·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.999i·5-s + (0.597 − 0.801i)7-s + 0.353·8-s + 0.707i·10-s − 0.426i·11-s + 0.877·13-s + (0.422 − 0.566i)14-s + 0.250·16-s + 1.08i·17-s + 0.499i·20-s − 0.301i·22-s + 1.25·23-s − 1.00·25-s + 0.620·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.45348 + 0.389391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45348 + 0.389391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (-1.58 + 2.12i)T \) |
| good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 4.47iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−10.82166167205864647846410754642, −10.20638875712043470804685854875, −8.743903792850516094798192297347, −7.83231015712028772804135153213, −6.90163214638191383753857212386, −6.21694042904162484682703684388, −5.08924272246594649996249701653, −3.88923450642458968560413928203, −3.17402805827084384692097579422, −1.59507422869635237095676200081,
1.40782062468486522491108836982, 2.76048153591015490199788523495, 4.19095656564932367032523326488, 5.05981138889256242548694141051, 5.68705620820396830090844243138, 6.88910330650222965235845248510, 7.973604828371737309816424357368, 8.809187685536511776585228873144, 9.525907604590774720085653223709, 10.79508999718165959044018978017
