| L(s) = 1 | + 3·3-s + 5·5-s − 28·7-s + 9·9-s + 24·11-s + 70·13-s + 15·15-s + 102·17-s − 20·19-s − 84·21-s − 72·23-s + 25·25-s + 27·27-s − 306·29-s − 136·31-s + 72·33-s − 140·35-s + 214·37-s + 210·39-s − 150·41-s + 292·43-s + 45·45-s − 72·47-s + 441·49-s + 306·51-s + 414·53-s + 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.657·11-s + 1.49·13-s + 0.258·15-s + 1.45·17-s − 0.241·19-s − 0.872·21-s − 0.652·23-s + 1/5·25-s + 0.192·27-s − 1.95·29-s − 0.787·31-s + 0.379·33-s − 0.676·35-s + 0.950·37-s + 0.862·39-s − 0.571·41-s + 1.03·43-s + 0.149·45-s − 0.223·47-s + 9/7·49-s + 0.840·51-s + 1.07·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.687049504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687049504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 306 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 418 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 480 T + p^{3} T^{2} \) |
| 73 | \( 1 - 434 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 - 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.527449188496564874647540494467, −9.041433134115375225506785035241, −8.037772773440825822037446396549, −7.04485613865871259914851622575, −6.17181619306944855329262544002, −5.59773583117081352948735848250, −3.70503380759648724341808334288, −3.59893736256981220306004362482, −2.15820728565203900782926144117, −0.878897593123218354206422579413,
0.878897593123218354206422579413, 2.15820728565203900782926144117, 3.59893736256981220306004362482, 3.70503380759648724341808334288, 5.59773583117081352948735848250, 6.17181619306944855329262544002, 7.04485613865871259914851622575, 8.037772773440825822037446396549, 9.041433134115375225506785035241, 9.527449188496564874647540494467
