| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 18-s + 20-s − 23-s − 24-s + 25-s − 27-s − 30-s + 32-s + 36-s + 40-s + 45-s − 46-s − 48-s + 49-s + 50-s − 2·53-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 18-s + 20-s − 23-s − 24-s + 25-s − 27-s − 30-s + 32-s + 36-s + 40-s + 45-s − 46-s − 48-s + 49-s + 50-s − 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.118954367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118954367\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.227572705339007908852963840543, −7.955503395821138780719523650357, −7.17688449312629682987870972997, −6.26585315633252284594024567413, −6.02764379137317124498425689362, −5.12969814837862498435389984774, −4.54685535210042580369619873603, −3.51733326428267753876245656928, −2.31932461532076265426808933270, −1.41328069906750323002042766527,
1.41328069906750323002042766527, 2.31932461532076265426808933270, 3.51733326428267753876245656928, 4.54685535210042580369619873603, 5.12969814837862498435389984774, 6.02764379137317124498425689362, 6.26585315633252284594024567413, 7.17688449312629682987870972997, 7.955503395821138780719523650357, 9.227572705339007908852963840543
