| L(s) = 1 | + 9·3-s − 25·5-s + 100·7-s + 81·9-s + 136·11-s + 82·13-s − 225·15-s + 358·17-s − 796·19-s + 900·21-s − 488·23-s + 625·25-s + 729·27-s + 7.46e3·29-s − 2.72e3·31-s + 1.22e3·33-s − 2.50e3·35-s + 7.79e3·37-s + 738·39-s + 1.82e4·41-s + 2.44e3·43-s − 2.02e3·45-s + 2.20e3·47-s − 6.80e3·49-s + 3.22e3·51-s + 1.01e4·53-s − 3.40e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.771·7-s + 1/3·9-s + 0.338·11-s + 0.134·13-s − 0.258·15-s + 0.300·17-s − 0.505·19-s + 0.445·21-s − 0.192·23-s + 1/5·25-s + 0.192·27-s + 1.64·29-s − 0.509·31-s + 0.195·33-s − 0.344·35-s + 0.935·37-s + 0.0776·39-s + 1.69·41-s + 0.201·43-s − 0.149·45-s + 0.145·47-s − 0.405·49-s + 0.173·51-s + 0.494·53-s − 0.151·55-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
| \(L(3)\) |
\(\approx\) |
\(2.672463754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.672463754\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 100 T + p^{5} T^{2} \) |
| 11 | \( 1 - 136 T + p^{5} T^{2} \) |
| 13 | \( 1 - 82 T + p^{5} T^{2} \) |
| 17 | \( 1 - 358 T + p^{5} T^{2} \) |
| 19 | \( 1 + 796 T + p^{5} T^{2} \) |
| 23 | \( 1 + 488 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7466 T + p^{5} T^{2} \) |
| 31 | \( 1 + 88 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 7794 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18234 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2444 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2200 T + p^{5} T^{2} \) |
| 53 | \( 1 - 10122 T + p^{5} T^{2} \) |
| 59 | \( 1 - 6776 T + p^{5} T^{2} \) |
| 61 | \( 1 - 23398 T + p^{5} T^{2} \) |
| 67 | \( 1 - 9676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 13728 T + p^{5} T^{2} \) |
| 73 | \( 1 + 27390 T + p^{5} T^{2} \) |
| 79 | \( 1 - 93288 T + p^{5} T^{2} \) |
| 83 | \( 1 - 23276 T + p^{5} T^{2} \) |
| 89 | \( 1 - 102354 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49502 T + p^{5} T^{2} \) |
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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
−11.28606625417661464491747026836, −10.33862762390882570269255085111, −9.166549619184943606368831192780, −8.274770361549990332029823521315, −7.52489063440953291489532269578, −6.27712163040541415018339259819, −4.80765375374777552288011940389, −3.82736743980083225381780753014, −2.42901319931203284294248601332, −0.995078154505204701934048142916,
0.995078154505204701934048142916, 2.42901319931203284294248601332, 3.82736743980083225381780753014, 4.80765375374777552288011940389, 6.27712163040541415018339259819, 7.52489063440953291489532269578, 8.274770361549990332029823521315, 9.166549619184943606368831192780, 10.33862762390882570269255085111, 11.28606625417661464491747026836
