| L(s) = 1 | + (−2 + 2i)2-s + (7 + 7i)3-s − 8i·4-s − 28·6-s + (−9 + 9i)7-s + (16 + 16i)8-s + 71i·9-s + (56 − 56i)12-s − 36i·14-s − 64·16-s + (−142 − 142i)18-s − 126·21-s + (67 + 67i)23-s + 224i·24-s + (−308 + 308i)27-s + (72 + 72i)28-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.34 + 1.34i)3-s − i·4-s − 1.90·6-s + (−0.485 + 0.485i)7-s + (0.707 + 0.707i)8-s + 2.62i·9-s + (1.34 − 1.34i)12-s − 0.687i·14-s − 16-s + (−1.85 − 1.85i)18-s − 1.30·21-s + (0.607 + 0.607i)23-s + 1.90i·24-s + (−2.19 + 2.19i)27-s + (0.485 + 0.485i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.414151 + 1.45787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414151 + 1.45787i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 2i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-7 - 7i)T + 27iT^{2} \) |
| 7 | \( 1 + (9 - 9i)T - 343iT^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3iT^{2} \) |
| 17 | \( 1 + 4.91e3iT^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + (-67 - 67i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 306iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4iT^{2} \) |
| 41 | \( 1 - 252T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-297 - 297i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-301 + 301i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 1.48e5iT^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 952T + 2.26e5T^{2} \) |
| 67 | \( 1 + (549 - 549i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5iT^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + (-77 - 77i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.38e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5iT^{2} \) |
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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
−14.22774463566472834301691485054, −13.28379639782719997707473172658, −11.19938464979710753428648221846, −10.02907197325396452097701247811, −9.400446945876812214536258651377, −8.563065667645598031474828155395, −7.53124519158098236271605183153, −5.70206066613869809970359024181, −4.26766288637287036372733008127, −2.56613981972683549730065298661,
0.989656648301615900332994597662, 2.50938608294968042267829940672, 3.66743269017678791364994085243, 6.76340070766615514269245152025, 7.50408053953268277628184208472, 8.627079628773992838951915190674, 9.363730313808020634950960989593, 10.73073796122239471903870329270, 12.25158570210643594130655285352, 12.84232479641795635739369832910
