L(s) = 1 | + (−1.39 + 0.257i)2-s + i·3-s + (1.86 − 0.717i)4-s − i·5-s + (−0.257 − 1.39i)6-s − 2.59·7-s + (−2.41 + 1.47i)8-s − 9-s + (0.257 + 1.39i)10-s + 1.75·11-s + (0.717 + 1.86i)12-s − 2.03·13-s + (3.61 − 0.670i)14-s + 15-s + (2.97 − 2.67i)16-s + 3.95i·17-s + ⋯ |
L(s) = 1 | + (−0.983 + 0.182i)2-s + 0.577i·3-s + (0.933 − 0.358i)4-s − 0.447i·5-s + (−0.105 − 0.567i)6-s − 0.982·7-s + (−0.852 + 0.522i)8-s − 0.333·9-s + (0.0815 + 0.439i)10-s + 0.528·11-s + (0.207 + 0.538i)12-s − 0.564·13-s + (0.965 − 0.179i)14-s + 0.258·15-s + (0.742 − 0.669i)16-s + 0.959i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.000974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2308935514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2308935514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.257i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-4.47 - 1.71i)T \) |
good | 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 3.95iT - 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 4.91iT - 31T^{2} \) |
| 37 | \( 1 + 0.890iT - 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.82T + 43T^{2} \) |
| 47 | \( 1 - 2.70iT - 47T^{2} \) |
| 53 | \( 1 - 2.16iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 7.39iT - 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 - 7.48iT - 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 - 15.8iT - 89T^{2} \) |
| 97 | \( 1 + 0.186iT - 97T^{2} \) |
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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
−9.695452877745648345185951102560, −9.363428715242161526023131245621, −8.594553067846265233620026933599, −7.63341080196482471605054586380, −6.85471190600115622376255353755, −5.93521374844825474760287616397, −5.19201984854263827156067695514, −3.82141380836968405923709025902, −2.92798162581173415531371561148, −1.47109708826024084498195667965,
0.13065446489841014820952220426, 1.58068801637653496050179120694, 2.89594715564875515447061793344, 3.39782945806444804713740373337, 5.17387406976178949456018974940, 6.27401852770338036309160812584, 7.08197956979513888097998687944, 7.28235880409716096044244409673, 8.474051703823363747190856419562, 9.306115244017411081430998804834