| L(s) = 1 | − 2·2-s − 5·3-s + 4·4-s + 5·5-s + 10·6-s + 7·7-s − 8·8-s − 2·9-s − 10·10-s − 39·11-s − 20·12-s − 43·13-s − 14·14-s − 25·15-s + 16·16-s + 69·17-s + 4·18-s + 92·19-s + 20·20-s − 35·21-s + 78·22-s + 23·23-s + 40·24-s + 25·25-s + 86·26-s + 145·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s + 1/2·4-s + 0.447·5-s + 0.680·6-s + 0.377·7-s − 0.353·8-s − 0.0740·9-s − 0.316·10-s − 1.06·11-s − 0.481·12-s − 0.917·13-s − 0.267·14-s − 0.430·15-s + 1/4·16-s + 0.984·17-s + 0.0523·18-s + 1.11·19-s + 0.223·20-s − 0.363·21-s + 0.755·22-s + 0.208·23-s + 0.340·24-s + 1/5·25-s + 0.648·26-s + 1.03·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 29 | \( 1 + 297 T + p^{3} T^{2} \) |
| 31 | \( 1 - 254 T + p^{3} T^{2} \) |
| 37 | \( 1 + 340 T + p^{3} T^{2} \) |
| 41 | \( 1 - 216 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 + 33 T + p^{3} T^{2} \) |
| 53 | \( 1 - 366 T + p^{3} T^{2} \) |
| 59 | \( 1 - 510 T + p^{3} T^{2} \) |
| 61 | \( 1 + 430 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 + 72 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 25 T + p^{3} T^{2} \) |
| 83 | \( 1 - 756 T + p^{3} T^{2} \) |
| 89 | \( 1 - 678 T + p^{3} T^{2} \) |
| 97 | \( 1 - 17 p T + p^{3} T^{2} \) |
| 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
−8.686883562925671918010302029053, −7.67485880254887006500391542292, −7.27084694444383875068548714079, −6.07680636602092249453613820776, −5.41507101739074393804091922868, −4.90961307804124845998194660283, −3.24590144144305730675996688243, −2.27088230984697079149679151830, −1.03748696962996715097823891815, 0,
1.03748696962996715097823891815, 2.27088230984697079149679151830, 3.24590144144305730675996688243, 4.90961307804124845998194660283, 5.41507101739074393804091922868, 6.07680636602092249453613820776, 7.27084694444383875068548714079, 7.67485880254887006500391542292, 8.686883562925671918010302029053
