| L(s) = 1 | + 3-s + 4.77·7-s + 9-s + 4.77·11-s + 13-s + 4.77·17-s + 6·19-s + 4.77·21-s − 6.77·23-s + 27-s + 2·29-s + 6·31-s + 4.77·33-s − 4.77·37-s + 39-s − 8.77·41-s − 8·43-s − 6·47-s + 15.7·49-s + 4.77·51-s + 4.77·53-s + 6·57-s − 3.54·59-s − 4.77·61-s + 4.77·63-s − 7.54·67-s − 6.77·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.80·7-s + 0.333·9-s + 1.43·11-s + 0.277·13-s + 1.15·17-s + 1.37·19-s + 1.04·21-s − 1.41·23-s + 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.830·33-s − 0.784·37-s + 0.160·39-s − 1.36·41-s − 1.21·43-s − 0.875·47-s + 2.25·49-s + 0.668·51-s + 0.655·53-s + 0.794·57-s − 0.461·59-s − 0.610·61-s + 0.601·63-s − 0.921·67-s − 0.815·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612574533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612574533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 4.77T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 4.77T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 + 7.54T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.402360782373956858328786957995, −7.87656392912426538060562426044, −7.21932988830234866506342368111, −6.27579344491861683473349057597, −5.37110869272206288758833574274, −4.64715454165012862846949776915, −3.86059844738386312988920356550, −3.05671997615694272464335478262, −1.64076033306419998682769096700, −1.33177132133500077326680996984,
1.33177132133500077326680996984, 1.64076033306419998682769096700, 3.05671997615694272464335478262, 3.86059844738386312988920356550, 4.64715454165012862846949776915, 5.37110869272206288758833574274, 6.27579344491861683473349057597, 7.21932988830234866506342368111, 7.87656392912426538060562426044, 8.402360782373956858328786957995
