L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s + 11-s − 4·13-s + 4·15-s + 4·17-s + 2·21-s + 4·23-s − 25-s − 4·27-s + 6·29-s − 10·31-s + 2·33-s + 2·35-s + 6·37-s − 8·39-s + 4·41-s + 12·43-s + 2·45-s + 10·47-s + 49-s + 8·51-s + 6·53-s + 2·55-s + 2·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.03·15-s + 0.970·17-s + 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.79·31-s + 0.348·33-s + 0.338·35-s + 0.986·37-s − 1.28·39-s + 0.624·41-s + 1.82·43-s + 0.298·45-s + 1.45·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 0.269·55-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.800768742\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.800768742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.318055770167249209020787556529, −7.50242612035465341765985002833, −7.15095220920519743848333951211, −5.89346819607895550605187044438, −5.47251856550031920805970764164, −4.46493777161093979419971313423, −3.61703962519106612479648235899, −2.61963574407090955942422458621, −2.22212830718058091155401288728, −1.05751961479654851181720694519,
1.05751961479654851181720694519, 2.22212830718058091155401288728, 2.61963574407090955942422458621, 3.61703962519106612479648235899, 4.46493777161093979419971313423, 5.47251856550031920805970764164, 5.89346819607895550605187044438, 7.15095220920519743848333951211, 7.50242612035465341765985002833, 8.318055770167249209020787556529