L(s) = 1 | + 1.41i·3-s + (1.22 + 1.87i)5-s + 2.64·7-s + 0.999·9-s − 3.46i·11-s + 2.44·13-s + (−2.64 + 1.73i)15-s + 4.89·17-s − 6.48·19-s + 3.74i·21-s + (−2 + 4.58i)25-s + 5.65i·27-s − 6·29-s + 4.89·33-s + (3.24 + 4.94i)35-s + ⋯ |
L(s) = 1 | + 0.816i·3-s + (0.547 + 0.836i)5-s + 0.999·7-s + 0.333·9-s − 1.04i·11-s + 0.679·13-s + (−0.683 + 0.447i)15-s + 1.18·17-s − 1.48·19-s + 0.816i·21-s + (−0.400 + 0.916i)25-s + 1.08i·27-s − 1.11·29-s + 0.852·33-s + (0.547 + 0.836i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57795 + 0.970971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57795 + 0.970971i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.22 - 1.87i)T \) |
| 7 | \( 1 - 2.64T \) |
good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 7.48iT - 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−10.91532993145059968693917743221, −10.21129077007459891533692131732, −9.266558772274155164827877681063, −8.347982656869909109069162582559, −7.39399605529486446347427078598, −6.17234452191215073281985829542, −5.41866201893728592316970859352, −4.18494849367790101095004057903, −3.26216081071959783349691340205, −1.71364936448650425198429667277,
1.34047748968225242711654782314, 2.05196576390523542014259789434, 4.10661955179690778677668997076, 5.01523587055777003121707448605, 6.03763820338653779372964590731, 7.08572601943665368229634578987, 7.989682009198410559985515891577, 8.642006082051091656129750836681, 9.766426521974510353203333181079, 10.52080392241116934419607118667