| L(s) = 1 | + 3-s − 5-s − 2·9-s + 3·11-s + 5·13-s − 15-s − 3·17-s + 2·19-s − 6·23-s + 25-s − 5·27-s − 3·29-s + 4·31-s + 3·33-s − 2·37-s + 5·39-s + 12·41-s + 10·43-s + 2·45-s − 9·47-s − 3·51-s − 12·53-s − 3·55-s + 2·57-s + 8·61-s − 5·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 1.38·13-s − 0.258·15-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.522·33-s − 0.328·37-s + 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.420·51-s − 1.64·53-s − 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.620·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.357291486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357291486\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.82538760213753, −15.66859709008256, −14.65790251484416, −14.39069309019990, −13.87772190262290, −13.32614984307172, −12.69985652159001, −12.00600120293010, −11.33648899011574, −11.18986481339413, −10.39675382045061, −9.437906843006411, −9.180241753634545, −8.518377372019239, −7.990596266729919, −7.519846979485403, −6.426373144115531, −6.233645812990390, −5.429521252229629, −4.415183282102728, −3.867488581503400, −3.360732566399367, −2.491466301278878, −1.660150292276173, −0.6509978804692007,
0.6509978804692007, 1.660150292276173, 2.491466301278878, 3.360732566399367, 3.867488581503400, 4.415183282102728, 5.429521252229629, 6.233645812990390, 6.426373144115531, 7.519846979485403, 7.990596266729919, 8.518377372019239, 9.180241753634545, 9.437906843006411, 10.39675382045061, 11.18986481339413, 11.33648899011574, 12.00600120293010, 12.69985652159001, 13.32614984307172, 13.87772190262290, 14.39069309019990, 14.65790251484416, 15.66859709008256, 15.82538760213753
