L(s) = 1 | + 2-s − 4-s − 2·5-s + 2·7-s − 3·8-s − 3·9-s − 2·10-s + 11-s − 5·13-s + 2·14-s − 16-s − 3·17-s − 3·18-s + 2·20-s + 22-s + 2·23-s − 25-s − 5·26-s − 2·28-s − 4·31-s + 5·32-s − 3·34-s − 4·35-s + 3·36-s − 37-s + 6·40-s + 12·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.755·7-s − 1.06·8-s − 9-s − 0.632·10-s + 0.301·11-s − 1.38·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s − 0.707·18-s + 0.447·20-s + 0.213·22-s + 0.417·23-s − 1/5·25-s − 0.980·26-s − 0.377·28-s − 0.718·31-s + 0.883·32-s − 0.514·34-s − 0.676·35-s + 1/2·36-s − 0.164·37-s + 0.948·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.134904088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.134904088\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + T \) |
| 109 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.436759829530981839389792779206, −7.70319283830800840043339822698, −7.08353164507191133148963624410, −5.92334957707777847918005519790, −5.37951573602703862728629730011, −4.48679642661361455667121616153, −4.14580220623246613658565604770, −3.07766484495031746213493806636, −2.28814593295268541772925963801, −0.52474791706451272902437294322,
0.52474791706451272902437294322, 2.28814593295268541772925963801, 3.07766484495031746213493806636, 4.14580220623246613658565604770, 4.48679642661361455667121616153, 5.37951573602703862728629730011, 5.92334957707777847918005519790, 7.08353164507191133148963624410, 7.70319283830800840043339822698, 8.436759829530981839389792779206