| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·13-s + 15-s + 6·17-s − 2·21-s + 8·23-s + 25-s − 27-s + 4·29-s − 2·35-s + 4·37-s + 4·39-s + 2·41-s + 8·43-s − 45-s − 8·47-s − 3·49-s − 6·51-s + 10·53-s + 6·59-s − 10·61-s + 2·63-s + 4·65-s − 6·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.338·35-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.840·51-s + 1.37·53-s + 0.781·59-s − 1.28·61-s + 0.251·63-s + 0.496·65-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346001448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346001448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.83197199224302, −12.29517812737821, −12.06004939281619, −11.51071570547648, −11.19334785829622, −10.62977736193222, −10.20779698481005, −9.764541885759815, −9.172101152103789, −8.746249548510469, −8.033345819395521, −7.675888343708020, −7.338642121354665, −6.799224915593955, −6.228683050397207, −5.533531546709086, −5.218643985548672, −4.684803544363172, −4.350771125611190, −3.593289222054729, −2.940683288178189, −2.548044476862702, −1.615407470456669, −1.061032964855739, −0.5116786870957751,
0.5116786870957751, 1.061032964855739, 1.615407470456669, 2.548044476862702, 2.940683288178189, 3.593289222054729, 4.350771125611190, 4.684803544363172, 5.218643985548672, 5.533531546709086, 6.228683050397207, 6.799224915593955, 7.338642121354665, 7.675888343708020, 8.033345819395521, 8.746249548510469, 9.172101152103789, 9.764541885759815, 10.20779698481005, 10.62977736193222, 11.19334785829622, 11.51071570547648, 12.06004939281619, 12.29517812737821, 12.83197199224302
