| L(s) = 1 | − 5-s − 3·7-s + 3·11-s − 13-s + 17-s − 6·19-s + 5·23-s + 25-s + 6·29-s + 2·31-s + 3·35-s + 7·37-s − 3·41-s − 8·43-s + 2·47-s + 2·49-s + 53-s − 3·55-s − 15·61-s + 65-s + 12·67-s − 5·71-s − 6·73-s − 9·77-s − 13·79-s − 12·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.904·11-s − 0.277·13-s + 0.242·17-s − 1.37·19-s + 1.04·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s + 0.507·35-s + 1.15·37-s − 0.468·41-s − 1.21·43-s + 0.291·47-s + 2/7·49-s + 0.137·53-s − 0.404·55-s − 1.92·61-s + 0.124·65-s + 1.46·67-s − 0.593·71-s − 0.702·73-s − 1.02·77-s − 1.46·79-s − 1.31·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.998700988118677797427524837007, −7.01896139648996386593439147672, −6.57339757327622466274175425287, −5.97395841223760559339064811663, −4.81314857615583609569093739118, −4.17298141849913517064014780570, −3.30871526953736058848123586225, −2.62014328905781798760071336665, −1.25658600735454635522531812385, 0,
1.25658600735454635522531812385, 2.62014328905781798760071336665, 3.30871526953736058848123586225, 4.17298141849913517064014780570, 4.81314857615583609569093739118, 5.97395841223760559339064811663, 6.57339757327622466274175425287, 7.01896139648996386593439147672, 7.998700988118677797427524837007
