| L(s) = 1 | + 3-s + 11-s − 13-s + 19-s + 23-s + 25-s − 27-s + 2·29-s + 33-s − 37-s − 39-s − 41-s − 2·47-s + 49-s − 53-s + 57-s − 61-s + 69-s + 75-s + 79-s − 81-s − 2·83-s + 2·87-s − 89-s − 101-s − 107-s − 111-s + ⋯ |
| L(s) = 1 | + 3-s + 11-s − 13-s + 19-s + 23-s + 25-s − 27-s + 2·29-s + 33-s − 37-s − 39-s − 41-s − 2·47-s + 49-s − 53-s + 57-s − 61-s + 69-s + 75-s + 79-s − 81-s − 2·83-s + 2·87-s − 89-s − 101-s − 107-s − 111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.583997911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583997911\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−9.416568165661064828325574891137, −8.729702169175044023077996316357, −8.128229193080369318838221062306, −7.12699438029582317522019527625, −6.59461948257335873781208059516, −5.29259814715282110751262168275, −4.54401707467350727789822750716, −3.29863113264769779121767674208, −2.80640427164256468295740918371, −1.44423586626459901136002168602,
1.44423586626459901136002168602, 2.80640427164256468295740918371, 3.29863113264769779121767674208, 4.54401707467350727789822750716, 5.29259814715282110751262168275, 6.59461948257335873781208059516, 7.12699438029582317522019527625, 8.128229193080369318838221062306, 8.729702169175044023077996316357, 9.416568165661064828325574891137
