| L(s) = 1 | − i·3-s + (−0.594 − 2.15i)5-s + 4.92i·7-s − 9-s + 1.38·11-s − i·13-s + (−2.15 + 0.594i)15-s + 0.195i·17-s + 4.92·21-s − 2.19i·23-s + (−4.29 + 2.56i)25-s + i·27-s − 7.49·29-s − 1.38i·33-s + (10.6 − 2.92i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.265 − 0.964i)5-s + 1.86i·7-s − 0.333·9-s + 0.417·11-s − 0.277i·13-s + (−0.556 + 0.153i)15-s + 0.0473i·17-s + 1.07·21-s − 0.457i·23-s + (−0.858 + 0.512i)25-s + 0.192i·27-s − 1.39·29-s − 0.240i·33-s + (1.79 − 0.494i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6996220867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6996220867\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.594 + 2.15i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.92iT - 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 17 | \( 1 - 0.195iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.19iT - 23T^{2} \) |
| 29 | \( 1 + 7.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.05iT - 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 - 3.49iT - 43T^{2} \) |
| 47 | \( 1 + 2.31iT - 47T^{2} \) |
| 53 | \( 1 - 6.05iT - 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.73iT - 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 - 6.01iT - 97T^{2} \) |
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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
−8.893698526295553346928433928805, −8.282467041943789831676503684030, −7.59147195828156285096528299323, −6.53369712835092520636660344567, −5.76405862802335056503149344765, −5.30215146582418971998098081571, −4.35462685482073938060939456675, −3.18405829302524749912872217475, −2.24152533918423283725350812202, −1.33932172236459579893969008226,
0.21652667433679680995345736458, 1.70991079803197642200967126100, 3.09211367868720781543694887488, 3.89978826935215881007557515321, 4.18117833344526592019013927147, 5.37338406598272500987179082166, 6.36186455359610473535279932933, 7.15751998324336030105429469376, 7.45644101123678489578505148758, 8.401885986163076181543424317755
