| L(s) = 1 | + i·2-s + i·3-s − 4-s + (0.806 + 2.08i)5-s − 6-s − i·8-s − 9-s + (−2.08 + 0.806i)10-s − 4.78·11-s − i·12-s − 3.17i·13-s + (−2.08 + 0.806i)15-s + 16-s − 5.22i·17-s − i·18-s − 3.17·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.360 + 0.932i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.659 + 0.255i)10-s − 1.44·11-s − 0.288i·12-s − 0.879i·13-s + (−0.538 + 0.208i)15-s + 0.250·16-s − 1.26i·17-s − 0.235i·18-s − 0.727·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2040327759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2040327759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.806 - 2.08i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 + 5.22iT - 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 + 7.17iT - 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 - 7.17iT - 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 11.5iT - 47T^{2} \) |
| 53 | \( 1 - 8.11iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 - 9.56iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 3.39iT - 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 - 1.27iT - 97T^{2} \) |
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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
−9.367692915930765998022180615532, −8.453414940465465619854570624040, −7.68683472854622129880143089742, −6.95473217599351483914478993247, −6.00319211185459689277505916971, −5.30182819762312333438316712092, −4.49836440968237360086329255152, −3.17373679532739755714162666024, −2.49401530361223631991477100296, −0.07628287618373447445220759771,
1.56445677607076047025220064614, 2.20998805239356689232886078334, 3.59072745456594282346403574202, 4.59518940668234035355814591122, 5.50484860265721607693289869576, 6.16665733617471200519924118786, 7.53889033863424725774076976872, 8.086295020341398999710860118149, 9.011009938987869649938701592523, 9.524784839029802552862597589304
