L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯ |
L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4789155944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4789155944\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.862262329947913428343529070066, −8.258538725477221090726449746241, −7.43927517609107944454842365043, −6.72363180263083440484828215114, −5.90167615767299762479450629027, −4.90290354737603484033044377537, −4.29092155912826760244323363959, −3.47490674582878888548143372136, −1.63069526623862256761644080483, −0.819001488182938398376713417081,
0.819001488182938398376713417081, 1.63069526623862256761644080483, 3.47490674582878888548143372136, 4.29092155912826760244323363959, 4.90290354737603484033044377537, 5.90167615767299762479450629027, 6.72363180263083440484828215114, 7.43927517609107944454842365043, 8.258538725477221090726449746241, 8.862262329947913428343529070066