| L(s) = 1 | + 3·3-s + 4·7-s + 6·9-s − 3·11-s + 6·13-s − 2·17-s − 19-s + 12·21-s + 9·27-s − 2·29-s − 10·31-s − 9·33-s − 8·37-s + 18·39-s + 5·41-s − 11·43-s + 2·47-s + 9·49-s − 6·51-s + 12·53-s − 3·57-s − 13·59-s + 8·61-s + 24·63-s + 3·67-s − 10·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.51·7-s + 2·9-s − 0.904·11-s + 1.66·13-s − 0.485·17-s − 0.229·19-s + 2.61·21-s + 1.73·27-s − 0.371·29-s − 1.79·31-s − 1.56·33-s − 1.31·37-s + 2.88·39-s + 0.780·41-s − 1.67·43-s + 0.291·47-s + 9/7·49-s − 0.840·51-s + 1.64·53-s − 0.397·57-s − 1.69·59-s + 1.02·61-s + 3.02·63-s + 0.366·67-s − 1.18·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.34164220084157, −12.99628000558333, −12.52671243009133, −11.78808795898790, −11.23157129778723, −10.86155720496877, −10.50635322381602, −9.930523565295381, −9.231429723610256, −8.816319328395302, −8.477965185859284, −8.232112779427486, −7.667761240401344, −7.252847179440312, −6.789076102980390, −5.931312148563202, −5.367958790485908, −4.957609869706802, −4.093475423348215, −3.955644730019425, −3.320406891149038, −2.694742233168878, −2.100248745628870, −1.623537035407646, −1.262120688458976, 0,
1.262120688458976, 1.623537035407646, 2.100248745628870, 2.694742233168878, 3.320406891149038, 3.955644730019425, 4.093475423348215, 4.957609869706802, 5.367958790485908, 5.931312148563202, 6.789076102980390, 7.252847179440312, 7.667761240401344, 8.232112779427486, 8.477965185859284, 8.816319328395302, 9.231429723610256, 9.930523565295381, 10.50635322381602, 10.86155720496877, 11.23157129778723, 11.78808795898790, 12.52671243009133, 12.99628000558333, 13.34164220084157
