| L(s) = 1 | − 2.54·3-s − 0.818·5-s + 2.13·7-s + 3.49·9-s − 0.530·11-s + 1.02·13-s + 2.08·15-s − 7.45·17-s − 0.396·19-s − 5.43·21-s − 4.32·25-s − 1.26·27-s − 8.23·29-s − 7.63·31-s + 1.35·33-s − 1.74·35-s − 1.74·37-s − 2.60·39-s + 8.78·41-s + 10.6·43-s − 2.86·45-s + 4.39·47-s − 2.45·49-s + 18.9·51-s + 5.42·53-s + 0.433·55-s + 1.00·57-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.366·5-s + 0.806·7-s + 1.16·9-s − 0.159·11-s + 0.283·13-s + 0.538·15-s − 1.80·17-s − 0.0908·19-s − 1.18·21-s − 0.865·25-s − 0.242·27-s − 1.52·29-s − 1.37·31-s + 0.235·33-s − 0.295·35-s − 0.287·37-s − 0.416·39-s + 1.37·41-s + 1.61·43-s − 0.426·45-s + 0.641·47-s − 0.350·49-s + 2.65·51-s + 0.745·53-s + 0.0585·55-s + 0.133·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6122944346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6122944346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.818T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 + 0.530T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 19 | \( 1 + 0.396T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 + 9.63T + 83T^{2} \) |
| 89 | \( 1 - 0.481T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−7.38633336699371893440681550568, −7.28539450317344860358619491921, −6.19316453336965670002866044273, −5.74943448214964548910097108973, −5.11552606128830218691468914200, −4.27996428706606217597851163920, −3.92833343289595879178957622974, −2.43425615661473001722233202150, −1.61470803360542685265041286214, −0.41527449865807725433507506385,
0.41527449865807725433507506385, 1.61470803360542685265041286214, 2.43425615661473001722233202150, 3.92833343289595879178957622974, 4.27996428706606217597851163920, 5.11552606128830218691468914200, 5.74943448214964548910097108973, 6.19316453336965670002866044273, 7.28539450317344860358619491921, 7.38633336699371893440681550568
