L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 6·11-s + 4·13-s + 16-s − 2·19-s − 20-s + 6·22-s + 3·23-s + 25-s + 4·26-s + 3·29-s − 8·31-s + 32-s − 4·37-s − 2·38-s − 40-s + 9·41-s − 7·43-s + 6·44-s + 3·46-s + 50-s + 4·52-s + 6·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1.27·22-s + 0.625·23-s + 1/5·25-s + 0.784·26-s + 0.557·29-s − 1.43·31-s + 0.176·32-s − 0.657·37-s − 0.324·38-s − 0.158·40-s + 1.40·41-s − 1.06·43-s + 0.904·44-s + 0.442·46-s + 0.141·50-s + 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.442659888\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.442659888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.438539401876684989258212691369, −7.44935149012820598471125798977, −6.71229899057828670651407729850, −6.26783616614327022024695196390, −5.39696469153382805762506100973, −4.43323482137427833285312312110, −3.80823904381667241835074198882, −3.27761662240906671499306546886, −1.94165700512029329412075874369, −1.00572145580708979387606425651,
1.00572145580708979387606425651, 1.94165700512029329412075874369, 3.27761662240906671499306546886, 3.80823904381667241835074198882, 4.43323482137427833285312312110, 5.39696469153382805762506100973, 6.26783616614327022024695196390, 6.71229899057828670651407729850, 7.44935149012820598471125798977, 8.438539401876684989258212691369