| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s + 5·11-s + 3·13-s + 3·15-s + 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s − 9·27-s + 9·29-s − 15·33-s + 35-s − 2·37-s − 9·39-s − 4·41-s − 10·43-s − 6·45-s − 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s − 18·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 2.61·33-s + 0.169·35-s − 0.328·37-s − 1.44·39-s − 0.624·41-s − 1.52·43-s − 0.894·45-s − 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.75067010729897, −13.05823655058933, −12.45151594260580, −12.06609052035680, −11.82982068074982, −11.37019927023812, −11.04726080689740, −10.31128612808392, −9.843709393336234, −9.689593157300355, −8.756061361671382, −8.368841824462629, −7.699521233270504, −7.002250914296449, −6.627636592849099, −6.326369912415466, −5.770666760503833, −5.243245670926244, −4.665132659894129, −4.157601092422263, −3.548186585613283, −3.170111219169885, −1.920619912919388, −1.232573414186136, −0.8568429529364635, 0,
0.8568429529364635, 1.232573414186136, 1.920619912919388, 3.170111219169885, 3.548186585613283, 4.157601092422263, 4.665132659894129, 5.243245670926244, 5.770666760503833, 6.326369912415466, 6.627636592849099, 7.002250914296449, 7.699521233270504, 8.368841824462629, 8.756061361671382, 9.689593157300355, 9.843709393336234, 10.31128612808392, 11.04726080689740, 11.37019927023812, 11.82982068074982, 12.06609052035680, 12.45151594260580, 13.05823655058933, 13.75067010729897
