| L(s) = 1 | + i·5-s − 2·7-s + 4i·11-s + 6·17-s − 4i·19-s − 4·23-s − 25-s + 6i·29-s − 10·31-s − 2i·35-s + 4i·37-s − 10·41-s + 4i·43-s − 4·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.755·7-s + 1.20i·11-s + 1.45·17-s − 0.917i·19-s − 0.834·23-s − 0.200·25-s + 1.11i·29-s − 1.79·31-s − 0.338i·35-s + 0.657i·37-s − 1.56·41-s + 0.609i·43-s − 0.583·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8479143869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8479143869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.821829431708892956964254494006, −9.255104980425820087005714032841, −8.138988632522686263166671829296, −7.24466066830627603973922874964, −6.76630783323006265674513399981, −5.72337206167562305029026070245, −4.85440100223836599974660812731, −3.69490182960377994125609730582, −2.90385070972141966395465254389, −1.63036692113692049996276112208,
0.33172000155736559573941842711, 1.81808407337671702396876871288, 3.36046617832412111145231734701, 3.77902288694413270098235946716, 5.30241617411372266717401701421, 5.81800693973327795464505440020, 6.67565302581894547538290429185, 7.86282324019680072946234411252, 8.289424434526591841867225924358, 9.354094253682609085681039595783
