| L(s) = 1 | + (−1.10 + 1.33i)3-s − 3.03·5-s − 2.60i·7-s + (−0.569 − 2.94i)9-s + 4.32·11-s − 6.76·13-s + (3.34 − 4.05i)15-s + 3.03·17-s + 5.50i·19-s + (3.47 + 2.87i)21-s + (4.32 − 2.07i)23-s + 4.21·25-s + (4.56 + 2.48i)27-s + 1.89i·29-s − 1.79·31-s + ⋯ |
| L(s) = 1 | + (−0.636 + 0.771i)3-s − 1.35·5-s − 0.984i·7-s + (−0.189 − 0.981i)9-s + 1.30·11-s − 1.87·13-s + (0.864 − 1.04i)15-s + 0.736·17-s + 1.26i·19-s + (0.759 + 0.626i)21-s + (0.901 − 0.433i)23-s + 0.843·25-s + (0.878 + 0.478i)27-s + 0.351i·29-s − 0.322·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7462464150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7462464150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.10 - 1.33i)T \) |
| 23 | \( 1 + (-4.32 + 2.07i)T \) |
| good | 5 | \( 1 + 3.03T + 5T^{2} \) |
| 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 6.76T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.50iT - 19T^{2} \) |
| 29 | \( 1 - 1.89iT - 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 + 3.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.23iT - 41T^{2} \) |
| 43 | \( 1 + 2.93iT - 43T^{2} \) |
| 47 | \( 1 - 9.23iT - 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 5.06iT - 59T^{2} \) |
| 61 | \( 1 - 13.6iT - 61T^{2} \) |
| 67 | \( 1 - 8.41iT - 67T^{2} \) |
| 71 | \( 1 - 9.10iT - 71T^{2} \) |
| 73 | \( 1 - 0.729T + 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−10.02811233577214334665995316187, −9.453193739489846558288017460657, −8.338205849256149233923351324655, −7.34923137856900527285502422028, −6.92272250081858833333369593003, −5.64233410194584841079640743476, −4.52380039555625528949789881361, −4.07296398575162880756385349686, −3.20405754688305021737470276266, −0.942358712532768689993491338415,
0.48815951878474226248416575402, 2.14900876488395806084381453564, 3.32577465686537722725103365422, 4.67052515946701762708734284091, 5.28820743424559095482623369376, 6.50755306815474578722673819760, 7.23374069624849706135634708627, 7.77046336493716839766713396893, 8.819530497358917992966276164247, 9.512077764029662016287962438138
