| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6863248302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6863248302\) |
| \(L(1)\) |
|
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
−8.988693197701184441539453449944, −7.85489627775899976091585541767, −6.96949715042200758995786713949, −6.51047741389978594109449949444, −5.61353232294910435880250336411, −4.92576250502726334552818606204, −4.38339880542815054191260007904, −3.34349081305354393697355840051, −2.84870252127014379935014793773, −0.63537227091119104819326554903,
0.63537227091119104819326554903, 2.84870252127014379935014793773, 3.34349081305354393697355840051, 4.38339880542815054191260007904, 4.92576250502726334552818606204, 5.61353232294910435880250336411, 6.51047741389978594109449949444, 6.96949715042200758995786713949, 7.85489627775899976091585541767, 8.988693197701184441539453449944
