| L(s) = 1 | + 2·3-s − 2·4-s + 3·5-s + 9-s − 4·12-s − 13-s + 6·15-s + 4·16-s + 6·17-s + 7·19-s − 6·20-s + 3·23-s + 4·25-s − 4·27-s − 9·29-s − 5·31-s − 2·36-s + 2·37-s − 2·39-s + 6·41-s − 43-s + 3·45-s − 3·47-s + 8·48-s + 12·51-s + 2·52-s − 9·53-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1.34·5-s + 1/3·9-s − 1.15·12-s − 0.277·13-s + 1.54·15-s + 16-s + 1.45·17-s + 1.60·19-s − 1.34·20-s + 0.625·23-s + 4/5·25-s − 0.769·27-s − 1.67·29-s − 0.898·31-s − 1/3·36-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.152·43-s + 0.447·45-s − 0.437·47-s + 1.15·48-s + 1.68·51-s + 0.277·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192809362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192809362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−10.03148147866603089067459950733, −9.513361358049510508727228598963, −9.166717133010691319523246651397, −8.059674511475772558890445124278, −7.35100084789034730201905858533, −5.69003044725444814418386644498, −5.29343454176495501876748811750, −3.73616092552496566051685235363, −2.86956267148131998918823072126, −1.46947183400268699699754331247,
1.46947183400268699699754331247, 2.86956267148131998918823072126, 3.73616092552496566051685235363, 5.29343454176495501876748811750, 5.69003044725444814418386644498, 7.35100084789034730201905858533, 8.059674511475772558890445124278, 9.166717133010691319523246651397, 9.513361358049510508727228598963, 10.03148147866603089067459950733
