| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s + 2·13-s + 16-s + 4·19-s − 4·22-s + 2·26-s − 10·29-s − 8·31-s + 32-s − 2·37-s + 4·38-s + 10·41-s − 12·43-s − 4·44-s − 7·49-s + 2·52-s + 6·53-s − 10·58-s − 12·59-s + 10·61-s − 8·62-s + 64-s + 12·67-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.917·19-s − 0.852·22-s + 0.392·26-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.328·37-s + 0.648·38-s + 1.56·41-s − 1.82·43-s − 0.603·44-s − 49-s + 0.277·52-s + 0.824·53-s − 1.31·58-s − 1.56·59-s + 1.28·61-s − 1.01·62-s + 1/8·64-s + 1.46·67-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64346374538691, −13.20482336184475, −12.82658723512006, −12.52182813314270, −11.74923919573272, −11.31920417668497, −10.94470587084485, −10.54318232508096, −9.832553681653365, −9.439380025726308, −8.901628897886886, −8.091344843083523, −7.852827146283918, −7.260875308578263, −6.830038577987491, −6.097208235212883, −5.591038655280054, −5.228564737825517, −4.789821058445910, −3.908406315750333, −3.512815223227513, −3.069160188255488, −2.169652428652925, −1.864394773046054, −0.8929551286967987, 0,
0.8929551286967987, 1.864394773046054, 2.169652428652925, 3.069160188255488, 3.512815223227513, 3.908406315750333, 4.789821058445910, 5.228564737825517, 5.591038655280054, 6.097208235212883, 6.830038577987491, 7.260875308578263, 7.852827146283918, 8.091344843083523, 8.901628897886886, 9.439380025726308, 9.832553681653365, 10.54318232508096, 10.94470587084485, 11.31920417668497, 11.74923919573272, 12.52182813314270, 12.82658723512006, 13.20482336184475, 13.64346374538691
