| L(s) = 1 | − 9·3-s + 25·5-s + 160·7-s + 81·9-s + 596·11-s − 122·13-s − 225·15-s − 1.07e3·17-s − 796·19-s − 1.44e3·21-s + 1.08e3·23-s + 625·25-s − 729·27-s + 46·29-s + 4.95e3·31-s − 5.36e3·33-s + 4.00e3·35-s − 6.11e3·37-s + 1.09e3·39-s − 6·41-s + 2.41e4·43-s + 2.02e3·45-s − 1.34e4·47-s + 8.79e3·49-s + 9.70e3·51-s + 2.05e4·53-s + 1.49e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.23·7-s + 1/3·9-s + 1.48·11-s − 0.200·13-s − 0.258·15-s − 0.904·17-s − 0.505·19-s − 0.712·21-s + 0.428·23-s + 1/5·25-s − 0.192·27-s + 0.0101·29-s + 0.925·31-s − 0.857·33-s + 0.551·35-s − 0.734·37-s + 0.115·39-s − 0.000557·41-s + 1.98·43-s + 0.149·45-s − 0.890·47-s + 0.523·49-s + 0.522·51-s + 1.00·53-s + 0.664·55-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
| \(L(3)\) |
\(\approx\) |
\(2.282171920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.282171920\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 160 T + p^{5} T^{2} \) |
| 11 | \( 1 - 596 T + p^{5} T^{2} \) |
| 13 | \( 1 + 122 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1078 T + p^{5} T^{2} \) |
| 19 | \( 1 + 796 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1088 T + p^{5} T^{2} \) |
| 29 | \( 1 - 46 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4952 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6114 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6 T + p^{5} T^{2} \) |
| 43 | \( 1 - 24116 T + p^{5} T^{2} \) |
| 47 | \( 1 + 13480 T + p^{5} T^{2} \) |
| 53 | \( 1 - 20598 T + p^{5} T^{2} \) |
| 59 | \( 1 - 46756 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17404 T + p^{5} T^{2} \) |
| 71 | \( 1 + 26568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 75450 T + p^{5} T^{2} \) |
| 79 | \( 1 + 50472 T + p^{5} T^{2} \) |
| 83 | \( 1 + 33236 T + p^{5} T^{2} \) |
| 89 | \( 1 - 133194 T + p^{5} T^{2} \) |
| 97 | \( 1 + 42878 T + p^{5} T^{2} \) |
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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
−11.34271300721522173805415488945, −10.48725504927703662950120089505, −9.287607071301285952975925160104, −8.436733573885988146792448482852, −7.08728592877654737569293856679, −6.20766622415685252188590893773, −4.99676912809566821986783277678, −4.09558676811677266237205024691, −2.11127621422059197424245410802, −0.986791568055470022194032034317,
0.986791568055470022194032034317, 2.11127621422059197424245410802, 4.09558676811677266237205024691, 4.99676912809566821986783277678, 6.20766622415685252188590893773, 7.08728592877654737569293856679, 8.436733573885988146792448482852, 9.287607071301285952975925160104, 10.48725504927703662950120089505, 11.34271300721522173805415488945
