| L(s) = 1 | + 4-s − 7-s + 11-s + 16-s + 17-s − 2·23-s + 25-s − 28-s + 29-s − 31-s − 37-s − 2·41-s + 44-s + 59-s − 61-s + 64-s + 68-s − 77-s − 83-s − 2·92-s + 100-s − 109-s − 112-s + 113-s + 116-s − 119-s + ⋯ |
| L(s) = 1 | + 4-s − 7-s + 11-s + 16-s + 17-s − 2·23-s + 25-s − 28-s + 29-s − 31-s − 37-s − 2·41-s + 44-s + 59-s − 61-s + 64-s + 68-s − 77-s − 83-s − 2·92-s + 100-s − 109-s − 112-s + 113-s + 116-s − 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 747 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 747 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.135323210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135323210\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 83 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.33619189965697019407377375734, −9.996682816904729453131360341286, −8.881971603448477992855244581955, −7.902307055893139797493718055635, −6.85452021958640261655613469567, −6.38848313890960112239641248512, −5.42245941148376174312048033401, −3.83720846535040555089503431946, −3.05645665634385429426392071870, −1.65050504201118594445174625452,
1.65050504201118594445174625452, 3.05645665634385429426392071870, 3.83720846535040555089503431946, 5.42245941148376174312048033401, 6.38848313890960112239641248512, 6.85452021958640261655613469567, 7.902307055893139797493718055635, 8.881971603448477992855244581955, 9.996682816904729453131360341286, 10.33619189965697019407377375734
