| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 11-s − 2·14-s + 16-s − 2·17-s + 6·19-s − 22-s − 4·23-s − 5·25-s + 2·28-s − 2·29-s − 32-s + 2·34-s + 6·37-s − 6·38-s + 2·41-s + 10·43-s + 44-s + 4·46-s − 3·49-s + 5·50-s + 8·53-s − 2·56-s + 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.301·11-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.213·22-s − 0.834·23-s − 25-s + 0.377·28-s − 0.371·29-s − 0.176·32-s + 0.342·34-s + 0.986·37-s − 0.973·38-s + 0.312·41-s + 1.52·43-s + 0.150·44-s + 0.589·46-s − 3/7·49-s + 0.707·50-s + 1.09·53-s − 0.267·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.809315309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.809315309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.15840777631442, −14.39847515286446, −14.06037930467371, −13.56742186578663, −12.77675528539544, −12.26751212446687, −11.53669795278855, −11.39946991126145, −10.81843765361787, −10.04848383738027, −9.619189065779126, −9.173142496327307, −8.486030761659383, −7.878694090074317, −7.589643988666651, −6.931435447292281, −6.183582165073984, −5.659150580503307, −5.048366250964957, −4.178702219878488, −3.729772587071437, −2.724274640153940, −2.119928999604725, −1.373199171443945, −0.5980732249232375,
0.5980732249232375, 1.373199171443945, 2.119928999604725, 2.724274640153940, 3.729772587071437, 4.178702219878488, 5.048366250964957, 5.659150580503307, 6.183582165073984, 6.931435447292281, 7.589643988666651, 7.878694090074317, 8.486030761659383, 9.173142496327307, 9.619189065779126, 10.04848383738027, 10.81843765361787, 11.39946991126145, 11.53669795278855, 12.26751212446687, 12.77675528539544, 13.56742186578663, 14.06037930467371, 14.39847515286446, 15.15840777631442
