| L(s) = 1 | + (−1 + i)2-s − i·3-s − 2i·4-s + 5-s + (1 + i)6-s − 3i·7-s + (2 + 2i)8-s + 2·9-s + (−1 + i)10-s + 2·11-s − 2·12-s + (−3 + 2i)13-s + (3 + 3i)14-s − i·15-s − 4·16-s + 3·17-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s − 0.577i·3-s − i·4-s + 0.447·5-s + (0.408 + 0.408i)6-s − 1.13i·7-s + (0.707 + 0.707i)8-s + 0.666·9-s + (−0.316 + 0.316i)10-s + 0.603·11-s − 0.577·12-s + (−0.832 + 0.554i)13-s + (0.801 + 0.801i)14-s − 0.258i·15-s − 16-s + 0.727·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.802096 - 0.0794232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802096 - 0.0794232i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−14.03138070170820377342404104863, −12.93652707664178297812045231949, −11.55168050213096256500942544377, −10.12002502784925675527084533184, −9.613508474892148105860213063569, −7.948701491373206449920631763674, −7.15030621521244837397206939435, −6.19223577176150942201911223395, −4.45607701323885764529889498638, −1.50989599007397932630514486544,
2.23083718496778732570938837681, 3.97335010471256464363848913478, 5.66492165023256605873583598016, 7.43213155263816955805373058027, 8.742668970694198924173312210402, 9.756467649995391558827101569606, 10.25760428819074643973910806183, 11.83154549957557831819446561218, 12.35409975707263402349255573602, 13.65256715401797432042833844266
