| L(s) = 1 | − 7-s − 2·11-s + 2·13-s − 4·17-s + 4·19-s + 6·23-s − 5·25-s − 2·29-s + 6·37-s − 8·41-s + 8·43-s + 4·47-s + 49-s − 6·53-s + 14·61-s − 4·67-s + 2·71-s − 2·73-s + 2·77-s + 4·79-s + 12·83-s − 2·91-s + 6·97-s + 12·101-s − 8·103-s + 6·107-s + 18·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s + 0.554·13-s − 0.970·17-s + 0.917·19-s + 1.25·23-s − 25-s − 0.371·29-s + 0.986·37-s − 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 1.79·61-s − 0.488·67-s + 0.237·71-s − 0.234·73-s + 0.227·77-s + 0.450·79-s + 1.31·83-s − 0.209·91-s + 0.609·97-s + 1.19·101-s − 0.788·103-s + 0.580·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628716762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628716762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.461724669872786901402316813867, −7.66713573880685231641557306283, −7.02626250778021512242656918408, −6.21787449399701061816318774055, −5.50172231446915643145331091937, −4.71077514128705501364254964394, −3.77987640746394375541111582265, −2.97784119788221167637879623474, −2.03842897638405211015926221067, −0.72220178526617033144485089127,
0.72220178526617033144485089127, 2.03842897638405211015926221067, 2.97784119788221167637879623474, 3.77987640746394375541111582265, 4.71077514128705501364254964394, 5.50172231446915643145331091937, 6.21787449399701061816318774055, 7.02626250778021512242656918408, 7.66713573880685231641557306283, 8.461724669872786901402316813867
