| L(s) = 1 | − i·3-s + i·7-s + 2·9-s + 5·11-s − i·13-s + 3i·17-s − 6·19-s + 21-s − 6i·23-s − 5i·27-s + 9·29-s − 5i·33-s + 6i·37-s − 39-s + 8·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s + 0.666·9-s + 1.50·11-s − 0.277i·13-s + 0.727i·17-s − 1.37·19-s + 0.218·21-s − 1.25i·23-s − 0.962i·27-s + 1.67·29-s − 0.870i·33-s + 0.986i·37-s − 0.160·39-s + 1.24·41-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
| \(L(1)\) |
\(\approx\) |
\(2.168632007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168632007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.621558628888131725996982183576, −8.119712333489477104089705370130, −7.09215291387972219635902400670, −6.34297594437784900886650720339, −6.13030168592212171090360501842, −4.53472011507871214885613983570, −4.22599508646504641531400825902, −2.91393626943632861314098449353, −1.88936907801025550579584649019, −0.953098794642534924671170037941,
0.988372600563698092601907186479, 2.10484361239347587111460760102, 3.48732707024485648818763934785, 4.14072960315629652661252324693, 4.69916865708491812884123838227, 5.79821322329938886017915722256, 6.75801729878116252681644865565, 7.11536328608990756207508642635, 8.203778047250675101885150001209, 9.063008527891968678602381740206
