| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4.44·11-s + 13-s − 2·17-s − 2.89·19-s + 2·21-s − 4.89·23-s − 27-s − 2·29-s + 6.89·31-s − 4.44·33-s − 1.10·37-s − 39-s + 8.44·41-s + 11.7·43-s + 9.34·47-s − 3·49-s + 2·51-s − 6.89·53-s + 2.89·57-s + 0.449·59-s + 4·61-s − 2·63-s − 11.7·67-s + 4.89·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 0.333·9-s + 1.34·11-s + 0.277·13-s − 0.485·17-s − 0.665·19-s + 0.436·21-s − 1.02·23-s − 0.192·27-s − 0.371·29-s + 1.23·31-s − 0.774·33-s − 0.181·37-s − 0.160·39-s + 1.31·41-s + 1.79·43-s + 1.36·47-s − 0.428·49-s + 0.280·51-s − 0.947·53-s + 0.383·57-s + 0.0585·59-s + 0.512·61-s − 0.251·63-s − 1.44·67-s + 0.589·69-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
| \(L(1)\) |
\(\approx\) |
\(1.335919049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335919049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.89T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 8.44T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.34T + 47T^{2} \) |
| 53 | \( 1 + 6.89T + 53T^{2} \) |
| 59 | \( 1 - 0.449T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.619747459627172900448876458267, −7.60364581499820669415890773887, −6.86565175881940893733457626207, −6.07658630330822518229829618741, −5.91737464022652709821872792028, −4.39411577773291075150496672414, −4.14402545471620771880848372295, −3.01690221458898146023126220616, −1.87920303644199059329922466950, −0.68933917017972104997714686333,
0.68933917017972104997714686333, 1.87920303644199059329922466950, 3.01690221458898146023126220616, 4.14402545471620771880848372295, 4.39411577773291075150496672414, 5.91737464022652709821872792028, 6.07658630330822518229829618741, 6.86565175881940893733457626207, 7.60364581499820669415890773887, 8.619747459627172900448876458267
