| L(s) = 1 | − 12·3-s − 25·5-s − 54·7-s − 99·9-s + 121·11-s − 540·13-s + 300·15-s + 340·17-s + 952·19-s + 648·21-s − 1.09e3·23-s + 625·25-s + 4.10e3·27-s − 62·29-s + 7.56e3·31-s − 1.45e3·33-s + 1.35e3·35-s − 9.18e3·37-s + 6.48e3·39-s − 6.81e3·41-s + 1.33e4·43-s + 2.47e3·45-s + 2.24e4·47-s − 1.38e4·49-s − 4.08e3·51-s + 1.96e4·53-s − 3.02e3·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s − 0.416·7-s − 0.407·9-s + 0.301·11-s − 0.886·13-s + 0.344·15-s + 0.285·17-s + 0.604·19-s + 0.320·21-s − 0.430·23-s + 1/5·25-s + 1.08·27-s − 0.0136·29-s + 1.41·31-s − 0.232·33-s + 0.186·35-s − 1.10·37-s + 0.682·39-s − 0.633·41-s + 1.09·43-s + 0.182·45-s + 1.48·47-s − 0.826·49-s − 0.219·51-s + 0.961·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 54 T + p^{5} T^{2} \) |
| 13 | \( 1 + 540 T + p^{5} T^{2} \) |
| 17 | \( 1 - 20 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 952 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1092 T + p^{5} T^{2} \) |
| 29 | \( 1 + 62 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7560 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9186 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6818 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13310 T + p^{5} T^{2} \) |
| 47 | \( 1 - 22420 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19654 T + p^{5} T^{2} \) |
| 59 | \( 1 + 48292 T + p^{5} T^{2} \) |
| 61 | \( 1 - 17530 T + p^{5} T^{2} \) |
| 67 | \( 1 - 35344 T + p^{5} T^{2} \) |
| 71 | \( 1 - 22912 T + p^{5} T^{2} \) |
| 73 | \( 1 - 47852 T + p^{5} T^{2} \) |
| 79 | \( 1 + 52396 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7890 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41958 T + p^{5} T^{2} \) |
| 97 | \( 1 + 37602 T + p^{5} T^{2} \) |
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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
−9.019577677285468485662836995593, −8.072198113466222306502601504105, −7.17004559320051432429081183931, −6.35258391628110922598360901098, −5.47710621337659164813365057440, −4.65424962831485825335309096948, −3.51800654043730597279596304988, −2.49995112888818036194355710561, −0.935075313101379459893891486611, 0,
0.935075313101379459893891486611, 2.49995112888818036194355710561, 3.51800654043730597279596304988, 4.65424962831485825335309096948, 5.47710621337659164813365057440, 6.35258391628110922598360901098, 7.17004559320051432429081183931, 8.072198113466222306502601504105, 9.019577677285468485662836995593
