| L(s) = 1 | + 3-s − 2·9-s + 2·11-s + 5·13-s + 4·17-s − 2·19-s + 23-s − 5·27-s − 3·29-s + 7·31-s + 2·33-s + 2·37-s + 5·39-s − 9·41-s + 4·43-s + 9·47-s − 7·49-s + 4·51-s + 6·53-s − 2·57-s + 2·61-s + 2·67-s + 69-s − 71-s − 73-s − 14·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.603·11-s + 1.38·13-s + 0.970·17-s − 0.458·19-s + 0.208·23-s − 0.962·27-s − 0.557·29-s + 1.25·31-s + 0.348·33-s + 0.328·37-s + 0.800·39-s − 1.40·41-s + 0.609·43-s + 1.31·47-s − 49-s + 0.560·51-s + 0.824·53-s − 0.264·57-s + 0.256·61-s + 0.244·67-s + 0.120·69-s − 0.118·71-s − 0.117·73-s − 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564001437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564001437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.435243997903484318374475159202, −7.77833414509059398402616514005, −6.86201711687664643235471721148, −6.07407266793959873473872207333, −5.56638186402414393744798053505, −4.43256685387981783740580101616, −3.61665393943710191096220547276, −3.04828722449270615275411826812, −1.95744040786654819114318229983, −0.896457012709257943054796121576,
0.896457012709257943054796121576, 1.95744040786654819114318229983, 3.04828722449270615275411826812, 3.61665393943710191096220547276, 4.43256685387981783740580101616, 5.56638186402414393744798053505, 6.07407266793959873473872207333, 6.86201711687664643235471721148, 7.77833414509059398402616514005, 8.435243997903484318374475159202
