L(s) = 1 | + i·3-s + (−1 − 2i)5-s + 5i·7-s − 9-s − 5·11-s − i·13-s + (2 − i)15-s − 3i·17-s − 4·19-s − 5·21-s + 5i·23-s + (−3 + 4i)25-s − i·27-s + 4·29-s − 5i·33-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.447 − 0.894i)5-s + 1.88i·7-s − 0.333·9-s − 1.50·11-s − 0.277i·13-s + (0.516 − 0.258i)15-s − 0.727i·17-s − 0.917·19-s − 1.09·21-s + 1.04i·23-s + (−0.600 + 0.800i)25-s − 0.192i·27-s + 0.742·29-s − 0.870i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6987404285\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6987404285\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 5iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 11T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−8.591047949550075871478787133855, −8.110090927072728382997584438213, −7.22991032648858796782306494284, −5.84245627309651084249190707967, −5.42663014605845049925119284201, −4.93516859292240993268529298400, −3.88979970121191682628409773550, −2.77828962518695630275832381903, −2.14176505761691054364407761983, −0.25789716590951574057422994658,
0.935039507705742142073608445260, 2.35301006351708017803347011602, 3.15540817771894792163069548177, 4.18546769875175989756241314618, 4.71513143668610859049671462006, 6.22408257724713854108591686867, 6.56008882336602238494155913746, 7.47621373781960526703507241229, 7.84380988510469127236053456005, 8.441201472848371988063470334681