| L(s) = 1 | + (−1.41 + 2.23i)7-s − 1.41i·11-s + 1.74·13-s + 4.47i·17-s − 1.74i·19-s − 3.16·23-s − 2.08i·29-s + 4.57i·31-s + 1.52i·37-s + 0.472·41-s + 0.472i·43-s + 2.47i·47-s + (−3.00 − 6.32i)49-s − 8.81·53-s − 1.52·59-s + ⋯ |
| L(s) = 1 | + (−0.534 + 0.845i)7-s − 0.426i·11-s + 0.484·13-s + 1.08i·17-s − 0.401i·19-s − 0.659·23-s − 0.386i·29-s + 0.821i·31-s + 0.251i·37-s + 0.0737·41-s + 0.0720i·43-s + 0.360i·47-s + (−0.428 − 0.903i)49-s − 1.21·53-s − 0.198·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4290562746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4290562746\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
| good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 1.74iT - 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 2.08iT - 29T^{2} \) |
| 31 | \( 1 - 4.57iT - 31T^{2} \) |
| 37 | \( 1 - 1.52iT - 37T^{2} \) |
| 41 | \( 1 - 0.472T + 41T^{2} \) |
| 43 | \( 1 - 0.472iT - 43T^{2} \) |
| 47 | \( 1 - 2.47iT - 47T^{2} \) |
| 53 | \( 1 + 8.81T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 3.49iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 7.73iT - 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 5.52iT - 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 0.412T + 97T^{2} \) |
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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
−8.383831172373274302881056654760, −7.86519324084442332849922610574, −6.77947175980435856686013594290, −6.22350374353559291348190341877, −5.70402721820204124687395914645, −4.85036495383333117943613105106, −3.89428578878789612516716498431, −3.21110313310574191516240827378, −2.35000587523982459537284689940, −1.35321899804944522687754089159,
0.11071704344403525076132651387, 1.24493739362405562460379287971, 2.34952562820501127388446956982, 3.31810339142596185220344436988, 3.99534396214010496096057030365, 4.72431639355251101814409665284, 5.60186476526268961753252640103, 6.38186974242209635718038586970, 7.00274294454580680487855469603, 7.64390609332922380134376548921
