L(s) = 1 | + 0.445·2-s − 3-s − 0.801·4-s − 5-s − 0.445·6-s − 0.801·8-s + 9-s − 0.445·10-s + 0.801·12-s + 15-s + 0.445·16-s − 1.24·17-s + 0.445·18-s − 1.80·19-s + 0.801·20-s + 0.445·23-s + 0.801·24-s + 25-s − 27-s + 0.445·30-s + 1.24·31-s + 32-s − 0.554·34-s − 0.801·36-s − 0.801·38-s + 0.801·40-s − 45-s + ⋯ |
L(s) = 1 | + 0.445·2-s − 3-s − 0.801·4-s − 5-s − 0.445·6-s − 0.801·8-s + 9-s − 0.445·10-s + 0.801·12-s + 15-s + 0.445·16-s − 1.24·17-s + 0.445·18-s − 1.80·19-s + 0.801·20-s + 0.445·23-s + 0.801·24-s + 25-s − 27-s + 0.445·30-s + 1.24·31-s + 32-s − 0.554·34-s − 0.801·36-s − 0.801·38-s + 0.801·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5441955640\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5441955640\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.24T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 - 0.445T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 - 1.80T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.761510908802385884605022533741, −8.604197371666475797603614921489, −7.39372863170780285403581424892, −6.67373804558687586965940834303, −5.93015445359599680135928889790, −4.99536281099735223934383526405, −4.25517079178473825863771014704, −3.97896192601681079354856351385, −2.50037299040305862237072227279, −0.66629022898785177965546375995,
0.66629022898785177965546375995, 2.50037299040305862237072227279, 3.97896192601681079354856351385, 4.25517079178473825863771014704, 4.99536281099735223934383526405, 5.93015445359599680135928889790, 6.67373804558687586965940834303, 7.39372863170780285403581424892, 8.604197371666475797603614921489, 8.761510908802385884605022533741