| L(s) = 1 | − i·3-s + (−2 + i)5-s − i·7-s − 9-s − 4·11-s + 2i·13-s + (1 + 2i)15-s + 6i·17-s − 6·19-s − 21-s + 2i·23-s + (3 − 4i)25-s + i·27-s + 6·29-s + 2·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s − 0.377i·7-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s + (0.258 + 0.516i)15-s + 1.45i·17-s − 1.37·19-s − 0.218·21-s + 0.417i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 1.11·29-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9383645981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9383645981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.284443263657071303800642119550, −7.78709175599320733188143519852, −7.05643779664937237974447119395, −6.39564653043328292349583891312, −5.59701362446903599351170961390, −4.41637977252001177270156144110, −3.88285571471713280032801835220, −2.77698854587098375223847251886, −1.93260249792191716124624186071, −0.41811313132571662303410385179,
0.74221561236188769776556404342, 2.59842619611779042103101380893, 3.04208717568522672933263043636, 4.41845411997967675558817394649, 4.69269676807223182706363432787, 5.56300785834979316518153890925, 6.45567261774950916125864063575, 7.50209084206200011746599559243, 8.082910385640537488895484195682, 8.665728545722395495658516393965
