L(s) = 1 | + 2.59i·3-s + (−1.73 − 1.41i)5-s + 3.38·7-s − 3.73·9-s + 1.75i·11-s − 6.21·13-s + (3.66 − 4.49i)15-s − 6.69i·17-s + (−1.34 − 4.14i)19-s + 8.78i·21-s − 5.86·23-s + (0.999 + 4.89i)25-s − 1.89i·27-s + 6.43i·29-s − 6.35·31-s + ⋯ |
L(s) = 1 | + 1.49i·3-s + (−0.774 − 0.632i)5-s + 1.27·7-s − 1.24·9-s + 0.528i·11-s − 1.72·13-s + (0.947 − 1.16i)15-s − 1.62i·17-s + (−0.308 − 0.951i)19-s + 1.91i·21-s − 1.22·23-s + (0.199 + 0.979i)25-s − 0.365i·27-s + 1.19i·29-s − 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1559307971\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1559307971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| 19 | \( 1 + (1.34 + 4.14i)T \) |
good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 13 | \( 1 + 6.21T + 13T^{2} \) |
| 17 | \( 1 + 6.69iT - 17T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 6.43iT - 29T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 + 1.66T + 37T^{2} \) |
| 41 | \( 1 + 8.78iT - 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.35T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 4.49iT - 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 3.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 + 6.43iT - 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−9.315686328079379617869988322904, −8.624974730755650858661372813977, −7.63383116126970468386057217679, −7.12709937087500780950206967101, −5.20769293423074464402845523757, −4.93698008903532747850711044180, −4.46886804063534644622460681506, −3.38411418616835474364420949464, −2.08348512953165644996620910475, −0.05765473919892732599701725933,
1.64074783182048112435051862488, 2.31640694460865085142858482555, 3.68428183502421444725976808885, 4.69888519007005834432074619628, 5.91745957960189331610263385214, 6.55297074199997774582506891498, 7.57429050007848325487245037457, 8.028297873483409447347252236809, 8.242169212534324684435357033574, 9.800577497423971964344958012828