L(s) = 1 | + i·3-s − i·7-s + 2·9-s − 3·11-s + i·13-s − 3i·17-s + 2·19-s + 21-s − 6i·23-s + 5i·27-s + 9·29-s − 8·31-s − 3i·33-s − 10i·37-s − 39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.377i·7-s + 0.666·9-s − 0.904·11-s + 0.277i·13-s − 0.727i·17-s + 0.458·19-s + 0.218·21-s − 1.25i·23-s + 0.962i·27-s + 1.67·29-s − 1.43·31-s − 0.522i·33-s − 1.64i·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.700421894\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.700421894\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−8.880631871836227922586445338244, −7.929934380071319571904516191256, −7.25245733959553561547882982114, −6.58497386198627546347670333010, −5.45425546604032510273831535691, −4.78544845855220300175518105108, −4.09308453566724708132214617670, −3.12436376633970318869610686510, −2.10470774340178935510859443156, −0.63231552227700797838597392081,
1.08554577552310083870792997118, 2.09695979403773195524816727514, 3.08573142890458198990343171127, 4.07567600208962810040399844629, 5.15594276701776437639526348154, 5.71338860406484897796926891192, 6.75029976839413447489882238920, 7.28673438779604457400750740585, 8.155540271893535838644369900654, 8.574901904478214231754256137834