| L(s) = 1 | + i·2-s − 0.302i·3-s − 4-s + 0.302·6-s − 4.60i·7-s − i·8-s + 2.90·9-s + 1.30·11-s + 0.302i·12-s − 2.30i·13-s + 4.60·14-s + 16-s + 6i·17-s + 2.90i·18-s − 2·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.174i·3-s − 0.5·4-s + 0.123·6-s − 1.74i·7-s − 0.353i·8-s + 0.969·9-s + 0.392·11-s + 0.0874i·12-s − 0.638i·13-s + 1.23·14-s + 0.250·16-s + 1.45i·17-s + 0.685i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.471759425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471759425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.302iT - 3T^{2} \) |
| 7 | \( 1 + 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 2.30iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.90iT - 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 + 0.908T + 41T^{2} \) |
| 43 | \( 1 + 6.60iT - 43T^{2} \) |
| 47 | \( 1 - 2.60iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 14.5iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.69iT - 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 17.2iT - 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−8.986845319187496416613010018571, −7.975997597319450431210831673884, −7.61022286175580840424156729173, −6.64808339352429282709096033937, −6.30087272990105668433948560233, −4.93171158631216274998438129249, −4.12974595855885795438674133190, −3.62079778364276613116429032460, −1.74670200477888196012967904422, −0.56774665187326436386963814195,
1.49857958825234531364959375963, 2.39197781983994159688185366103, 3.37000955572305406815820992691, 4.43809123102382197185587370236, 5.19516768060739279086933940796, 6.02735618369702835618610925151, 7.04342717995339147413024972463, 7.939258702848124228975144676821, 9.107242808474755464919339876143, 9.261035256779632435098519895436
