| L(s) = 1 | − 3-s − 2·9-s + 11-s − 2·13-s + 3·17-s − 3·19-s + 6·23-s + 5·27-s − 2·29-s + 2·31-s − 33-s + 4·37-s + 2·39-s − 3·41-s + 4·43-s − 6·47-s − 7·49-s − 3·51-s − 10·53-s + 3·57-s − 12·59-s − 12·61-s + 67-s − 6·69-s + 10·71-s + 73-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s − 0.554·13-s + 0.727·17-s − 0.688·19-s + 1.25·23-s + 0.962·27-s − 0.371·29-s + 0.359·31-s − 0.174·33-s + 0.657·37-s + 0.320·39-s − 0.468·41-s + 0.609·43-s − 0.875·47-s − 49-s − 0.420·51-s − 1.37·53-s + 0.397·57-s − 1.56·59-s − 1.53·61-s + 0.122·67-s − 0.722·69-s + 1.18·71-s + 0.117·73-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.223317273227340821933022445765, −7.59475580632656013816271748930, −6.59151966463464673964477053068, −6.12272624823240236669280480402, −5.16148678678551440880339510867, −4.64865647835609028571814005414, −3.43754551399960416633017373769, −2.65431138797723541397570093834, −1.33970185318465942188202813950, 0,
1.33970185318465942188202813950, 2.65431138797723541397570093834, 3.43754551399960416633017373769, 4.64865647835609028571814005414, 5.16148678678551440880339510867, 6.12272624823240236669280480402, 6.59151966463464673964477053068, 7.59475580632656013816271748930, 8.223317273227340821933022445765
