| L(s) = 1 | − 3-s − 5-s − 2·9-s + 11-s − 5·13-s + 15-s + 17-s + 6·19-s + 4·23-s + 25-s + 5·27-s + 3·29-s − 2·31-s − 33-s + 8·37-s + 5·39-s − 10·41-s + 2·43-s + 2·45-s + 7·47-s − 51-s − 2·53-s − 55-s − 6·57-s − 14·59-s − 8·61-s + 5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.301·11-s − 1.38·13-s + 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.359·31-s − 0.174·33-s + 1.31·37-s + 0.800·39-s − 1.56·41-s + 0.304·43-s + 0.298·45-s + 1.02·47-s − 0.140·51-s − 0.274·53-s − 0.134·55-s − 0.794·57-s − 1.82·59-s − 1.02·61-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.893936216873012795997598706111, −7.43833913647487274623928797033, −6.64369485395560191067557261252, −5.80358695937401583052128456079, −5.07114006116263062621462134438, −4.51620841570391596151335172237, −3.27771955812357426943631655909, −2.68369309862857984360252558725, −1.19869311136651987114745645340, 0,
1.19869311136651987114745645340, 2.68369309862857984360252558725, 3.27771955812357426943631655909, 4.51620841570391596151335172237, 5.07114006116263062621462134438, 5.80358695937401583052128456079, 6.64369485395560191067557261252, 7.43833913647487274623928797033, 7.893936216873012795997598706111
