| L(s) = 1 | + (1.95 + 1.08i)5-s + (−2.37 + 1.16i)7-s + 3.74i·11-s − 0.841·13-s − 3.36i·17-s + 4.55i·19-s − 7.64·23-s + (2.64 + 4.24i)25-s + 1.41i·29-s − 0.979i·31-s + (−5.90 − 0.302i)35-s + 2.32i·37-s − 10.3·41-s + 10.8i·43-s − 7.91i·47-s + ⋯ |
| L(s) = 1 | + (0.874 + 0.485i)5-s + (−0.898 + 0.439i)7-s + 1.12i·11-s − 0.233·13-s − 0.814i·17-s + 1.04i·19-s − 1.59·23-s + (0.529 + 0.848i)25-s + 0.262i·29-s − 0.175i·31-s + (−0.998 − 0.0511i)35-s + 0.382i·37-s − 1.61·41-s + 1.64i·43-s − 1.15i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185739304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185739304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.95 - 1.08i)T \) |
| 7 | \( 1 + (2.37 - 1.16i)T \) |
| good | 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 0.841T + 13T^{2} \) |
| 17 | \( 1 + 3.36iT - 17T^{2} \) |
| 19 | \( 1 - 4.55iT - 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 0.979iT - 31T^{2} \) |
| 37 | \( 1 - 2.32iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 7.91iT - 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.14iT - 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.951233735976287230036660390263, −9.463183208955358440452897041999, −8.401251336071403985131261586620, −7.33828416545970281815931060885, −6.62301291842177973956480300088, −5.88416813986602153750184940336, −5.02613000353631826071506308108, −3.77814742756227019016096287374, −2.69161651052109277193684847416, −1.81657778943203841402390537436,
0.46608902697854590629902737597, 1.98298992952777797434967355129, 3.18984076459147962456233914900, 4.16047482534188536451616274308, 5.33416423079307254730719500180, 6.11392948585495841704489530407, 6.70174976246785441109930821140, 7.88665399446905674790631074970, 8.753248193317307261141079205717, 9.375326423562119884944446925944
