L(s) = 1 | + 1.24·3-s + 4-s + 0.445·5-s + 0.554·9-s − 11-s + 1.24·12-s + 0.554·15-s + 16-s + 0.445·20-s − 1.80·23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s − 1.24·33-s + 0.554·36-s − 1.24·37-s − 44-s + 0.246·45-s + 1.80·47-s + 1.24·48-s + 49-s − 0.445·53-s − 0.445·55-s − 1.24·59-s + 0.554·60-s + 64-s − 2·67-s + ⋯ |
L(s) = 1 | + 1.24·3-s + 4-s + 0.445·5-s + 0.554·9-s − 11-s + 1.24·12-s + 0.554·15-s + 16-s + 0.445·20-s − 1.80·23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s − 1.24·33-s + 0.554·36-s − 1.24·37-s − 44-s + 0.246·45-s + 1.80·47-s + 1.24·48-s + 49-s − 0.445·53-s − 0.445·55-s − 1.24·59-s + 0.554·60-s + 64-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.111193160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.111193160\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 + 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + 1.24T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.80T + T^{2} \) |
| 97 | \( 1 - 0.445T + T^{2} \) |
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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
−9.408933157271934198136588809462, −8.501014710461866340974999531889, −7.85243625619789117458526318481, −7.36288778847386768586868736696, −6.19101814870035695047020715233, −5.65079930615928471228446956144, −4.29704197947666420624477345383, −3.22677663520357625486768732187, −2.48892331493762691856556133623, −1.82197688265522102554459922474,
1.82197688265522102554459922474, 2.48892331493762691856556133623, 3.22677663520357625486768732187, 4.29704197947666420624477345383, 5.65079930615928471228446956144, 6.19101814870035695047020715233, 7.36288778847386768586868736696, 7.85243625619789117458526318481, 8.501014710461866340974999531889, 9.408933157271934198136588809462