| L(s) = 1 | + 9·3-s − 25·5-s − 108·7-s + 81·9-s + 8·11-s + 162·13-s − 225·15-s − 714·17-s + 532·19-s − 972·21-s + 4.58e3·23-s + 625·25-s + 729·27-s + 938·29-s + 8.36e3·31-s + 72·33-s + 2.70e3·35-s + 1.09e3·37-s + 1.45e3·39-s − 1.12e4·41-s + 7.69e3·43-s − 2.02e3·45-s + 1.36e4·47-s − 5.14e3·49-s − 6.42e3·51-s + 1.90e4·53-s − 200·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.833·7-s + 1/3·9-s + 0.0199·11-s + 0.265·13-s − 0.258·15-s − 0.599·17-s + 0.338·19-s − 0.480·21-s + 1.80·23-s + 1/5·25-s + 0.192·27-s + 0.207·29-s + 1.56·31-s + 0.0115·33-s + 0.372·35-s + 0.130·37-s + 0.153·39-s − 1.04·41-s + 0.634·43-s − 0.149·45-s + 0.900·47-s − 0.306·49-s − 0.345·51-s + 0.931·53-s − 0.00891·55-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
| \(L(3)\) |
\(\approx\) |
\(2.080499304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080499304\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 108 T + p^{5} T^{2} \) |
| 11 | \( 1 - 8 T + p^{5} T^{2} \) |
| 13 | \( 1 - 162 T + p^{5} T^{2} \) |
| 17 | \( 1 + 42 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 28 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 4584 T + p^{5} T^{2} \) |
| 29 | \( 1 - 938 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 - 1090 T + p^{5} T^{2} \) |
| 41 | \( 1 + 11238 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7692 T + p^{5} T^{2} \) |
| 47 | \( 1 - 13640 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19050 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18936 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1978 T + p^{5} T^{2} \) |
| 67 | \( 1 + 44212 T + p^{5} T^{2} \) |
| 71 | \( 1 - 59744 T + p^{5} T^{2} \) |
| 73 | \( 1 - 56994 T + p^{5} T^{2} \) |
| 79 | \( 1 - 15128 T + p^{5} T^{2} \) |
| 83 | \( 1 - 21996 T + p^{5} T^{2} \) |
| 89 | \( 1 - 14066 T + p^{5} T^{2} \) |
| 97 | \( 1 - 75938 T + p^{5} T^{2} \) |
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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
−11.25326378850860671701257847979, −10.22433212880407218330197357862, −9.221371068353138537763174334631, −8.442707150032737160991820221751, −7.25917904543803825400758216124, −6.41083558021756526388905759431, −4.87276068389360221493870549899, −3.62317098884160883129512754470, −2.64117406869078812213251790094, −0.838809838248433351469118311609,
0.838809838248433351469118311609, 2.64117406869078812213251790094, 3.62317098884160883129512754470, 4.87276068389360221493870549899, 6.41083558021756526388905759431, 7.25917904543803825400758216124, 8.442707150032737160991820221751, 9.221371068353138537763174334631, 10.22433212880407218330197357862, 11.25326378850860671701257847979
