| L(s) = 1 | + (−1 + i)3-s + (1 − i)7-s + i·9-s + 4·11-s + (−3 − 3i)13-s + (3 + 3i)17-s − 6i·19-s + 2i·21-s + (3 + 3i)23-s + (−4 − 4i)27-s − 2·29-s − 6i·31-s + (−4 + 4i)33-s + (−3 + 3i)37-s + 6·39-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.377 − 0.377i)7-s + 0.333i·9-s + 1.20·11-s + (−0.832 − 0.832i)13-s + (0.727 + 0.727i)17-s − 1.37i·19-s + 0.436i·21-s + (0.625 + 0.625i)23-s + (−0.769 − 0.769i)27-s − 0.371·29-s − 1.07i·31-s + (−0.696 + 0.696i)33-s + (−0.493 + 0.493i)37-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.509645793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.509645793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (-3 - 3i)T + 23iT^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9 + 9i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + (-9 - 9i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-3 + 3i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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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
−9.533014053654160072655803997924, −8.779328227780187777344334953433, −7.67933377654365724977655464813, −7.17467628510064910004177427988, −5.98821861704053599551446532266, −5.29255845273448946373582166915, −4.48253475421756990229897947276, −3.68069322451626743985511515298, −2.35349285089622839385830368836, −0.866450618518392576208124893054,
0.983405959453079691104987812441, 2.01597518886371727199565323716, 3.41250153430051590262418356790, 4.41173621166901332633662406709, 5.42856529628178976286610297630, 6.17265562401493949478572748374, 6.97740468308658975810006489468, 7.53445006873851160867671529588, 8.719766673044612478918708286302, 9.305393178039716966822555472399
