| L(s) = 1 | − 9·3-s + 25·5-s − 56·7-s + 81·9-s − 156·11-s + 350·13-s − 225·15-s + 786·17-s − 740·19-s + 504·21-s − 2.37e3·23-s + 625·25-s − 729·27-s + 2.57e3·29-s + 4.57e3·31-s + 1.40e3·33-s − 1.40e3·35-s − 1.22e4·37-s − 3.15e3·39-s − 1.02e4·41-s + 1.60e4·43-s + 2.02e3·45-s − 864·47-s − 1.36e4·49-s − 7.07e3·51-s − 1.76e4·53-s − 3.90e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.431·7-s + 1/3·9-s − 0.388·11-s + 0.574·13-s − 0.258·15-s + 0.659·17-s − 0.470·19-s + 0.249·21-s − 0.936·23-s + 1/5·25-s − 0.192·27-s + 0.568·29-s + 0.855·31-s + 0.224·33-s − 0.193·35-s − 1.46·37-s − 0.331·39-s − 0.950·41-s + 1.32·43-s + 0.149·45-s − 0.0570·47-s − 0.813·49-s − 0.380·51-s − 0.863·53-s − 0.173·55-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 | \( 1 + 8 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 156 T + p^{5} T^{2} \) |
| 13 | \( 1 - 350 T + p^{5} T^{2} \) |
| 17 | \( 1 - 786 T + p^{5} T^{2} \) |
| 19 | \( 1 + 740 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2376 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4576 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12202 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16084 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17658 T + p^{5} T^{2} \) |
| 59 | \( 1 + 48684 T + p^{5} T^{2} \) |
| 61 | \( 1 + 33778 T + p^{5} T^{2} \) |
| 67 | \( 1 + 3524 T + p^{5} T^{2} \) |
| 71 | \( 1 + 38280 T + p^{5} T^{2} \) |
| 73 | \( 1 + 79702 T + p^{5} T^{2} \) |
| 79 | \( 1 + 99248 T + p^{5} T^{2} \) |
| 83 | \( 1 - 22284 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94650 T + p^{5} T^{2} \) |
| 97 | \( 1 - 9122 T + p^{5} T^{2} \) |
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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
−10.60808385906482659964256224360, −10.06548086632608812739585821094, −8.900566590076440848167445644901, −7.75823394506011029301260748519, −6.48946407836431391956933423565, −5.77818643155782558558420546555, −4.55331102895050284908003443631, −3.12073291557906327768627290428, −1.53895905774983190393173518118, 0,
1.53895905774983190393173518118, 3.12073291557906327768627290428, 4.55331102895050284908003443631, 5.77818643155782558558420546555, 6.48946407836431391956933423565, 7.75823394506011029301260748519, 8.900566590076440848167445644901, 10.06548086632608812739585821094, 10.60808385906482659964256224360
