| L(s) = 1 | − 3-s − 4-s + 7-s + 9-s + 12-s − 2·13-s + 16-s − 21-s + 25-s − 27-s − 28-s + 2·31-s − 36-s + 2·37-s + 2·39-s − 48-s + 49-s + 2·52-s + 2·61-s + 63-s − 64-s − 67-s − 75-s + 81-s + 84-s − 2·91-s − 2·93-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 7-s + 9-s + 12-s − 2·13-s + 16-s − 21-s + 25-s − 27-s − 28-s + 2·31-s − 36-s + 2·37-s + 2·39-s − 48-s + 49-s + 2·52-s + 2·61-s + 63-s − 64-s − 67-s − 75-s + 81-s + 84-s − 2·91-s − 2·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6561713271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6561713271\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 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
−9.939414379568968046291860265836, −9.054393190851836512138930196487, −8.021938775330449774571110484318, −7.44001426963983250232352728695, −6.39161208800599632228475441890, −5.29164844630623691948399419772, −4.79375916162197586948637722520, −4.24337908416153790588491171544, −2.55005962319756003614880891509, −0.947789898283743219229300410811,
0.947789898283743219229300410811, 2.55005962319756003614880891509, 4.24337908416153790588491171544, 4.79375916162197586948637722520, 5.29164844630623691948399419772, 6.39161208800599632228475441890, 7.44001426963983250232352728695, 8.021938775330449774571110484318, 9.054393190851836512138930196487, 9.939414379568968046291860265836
