| L(s) = 1 | − 3-s − 3.46·5-s + 9-s − 1.46·11-s − 2·13-s + 3.46·15-s − 0.535·17-s − 6.92·19-s − 1.46·23-s + 6.99·25-s − 27-s − 4.92·29-s + 10.9·31-s + 1.46·33-s − 2·37-s + 2·39-s − 11.4·41-s − 8·43-s − 3.46·45-s − 10.9·47-s + 0.535·51-s − 2·53-s + 5.07·55-s + 6.92·57-s + 1.07·59-s + 8.92·61-s + 6.92·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.54·5-s + 0.333·9-s − 0.441·11-s − 0.554·13-s + 0.894·15-s − 0.129·17-s − 1.58·19-s − 0.305·23-s + 1.39·25-s − 0.192·27-s − 0.915·29-s + 1.96·31-s + 0.254·33-s − 0.328·37-s + 0.320·39-s − 1.79·41-s − 1.21·43-s − 0.516·45-s − 1.59·47-s + 0.0750·51-s − 0.274·53-s + 0.683·55-s + 0.917·57-s + 0.139·59-s + 1.14·61-s + 0.859·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3547934479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3547934479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−8.270091430142331373694486550974, −7.62327723976503701743025320193, −6.81988881294969734229411687633, −6.32238596233301334453419553162, −5.12936950895129047366890772971, −4.61486582547975071262776718261, −3.88934075823611853422117487540, −3.04441679227922769430397884341, −1.86137984169800998788670805297, −0.32593782165191865726222136075,
0.32593782165191865726222136075, 1.86137984169800998788670805297, 3.04441679227922769430397884341, 3.88934075823611853422117487540, 4.61486582547975071262776718261, 5.12936950895129047366890772971, 6.32238596233301334453419553162, 6.81988881294969734229411687633, 7.62327723976503701743025320193, 8.270091430142331373694486550974
