| L(s) = 1 | − 9·3-s + 25·5-s − 32·7-s + 81·9-s − 12·11-s − 154·13-s − 225·15-s − 918·17-s + 1.06e3·19-s + 288·21-s + 4.22e3·23-s + 625·25-s − 729·27-s − 7.89e3·29-s − 5.19e3·31-s + 108·33-s − 800·35-s + 1.63e4·37-s + 1.38e3·39-s + 3.64e3·41-s − 1.51e4·43-s + 2.02e3·45-s − 2.35e4·47-s − 1.57e4·49-s + 8.26e3·51-s − 1.60e4·53-s − 300·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.246·7-s + 1/3·9-s − 0.0299·11-s − 0.252·13-s − 0.258·15-s − 0.770·17-s + 0.673·19-s + 0.142·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.74·29-s − 0.970·31-s + 0.0172·33-s − 0.110·35-s + 1.96·37-s + 0.145·39-s + 0.338·41-s − 1.24·43-s + 0.149·45-s − 1.55·47-s − 0.939·49-s + 0.444·51-s − 0.786·53-s − 0.0133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 32 T + p^{5} T^{2} \) |
| 11 | \( 1 + 12 T + p^{5} T^{2} \) |
| 13 | \( 1 + 154 T + p^{5} T^{2} \) |
| 17 | \( 1 + 54 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4224 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7890 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5192 T + p^{5} T^{2} \) |
| 37 | \( 1 - 16382 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3642 T + p^{5} T^{2} \) |
| 43 | \( 1 + 15116 T + p^{5} T^{2} \) |
| 47 | \( 1 + 23592 T + p^{5} T^{2} \) |
| 53 | \( 1 + 16074 T + p^{5} T^{2} \) |
| 59 | \( 1 - 14340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 47938 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33092 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51912 T + p^{5} T^{2} \) |
| 73 | \( 1 - 12026 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25160 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35796 T + p^{5} T^{2} \) |
| 89 | \( 1 + 75510 T + p^{5} T^{2} \) |
| 97 | \( 1 + 44158 T + p^{5} T^{2} \) |
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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
−10.97450714323315550692237227634, −9.748819433012787805921874371675, −9.098448081384412120265154461942, −7.61792609274342740879734572360, −6.66325670930641052195967482778, −5.62503321630122033192567934310, −4.62930261885043226387119956144, −3.09057428659590158490409349790, −1.55095139479547155645914505640, 0,
1.55095139479547155645914505640, 3.09057428659590158490409349790, 4.62930261885043226387119956144, 5.62503321630122033192567934310, 6.66325670930641052195967482778, 7.61792609274342740879734572360, 9.098448081384412120265154461942, 9.748819433012787805921874371675, 10.97450714323315550692237227634
