| L(s) = 1 | + 4·11-s + 2·13-s + 2·17-s + 4·19-s + 8·23-s − 6·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s − 7·49-s − 2·53-s + 4·59-s − 2·61-s − 4·67-s + 8·71-s − 10·73-s + 8·79-s + 4·83-s + 6·89-s − 2·97-s + 18·101-s + 16·103-s + 12·107-s − 2·109-s + 18·113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1.11·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s + 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s − 0.203·97-s + 1.79·101-s + 1.57·103-s + 1.16·107-s − 0.191·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227239783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227239783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.825846092924041988502122678303, −7.62196275802298861591436072358, −7.18888718297647721192763594914, −6.30978253779542156728101297956, −5.57054830111212428433722417563, −4.79739013158573068367880494367, −3.70365087696193807260970500510, −3.25678719031186811580504077978, −1.85007062551041405309421470085, −0.935452235825979143826970020046,
0.935452235825979143826970020046, 1.85007062551041405309421470085, 3.25678719031186811580504077978, 3.70365087696193807260970500510, 4.79739013158573068367880494367, 5.57054830111212428433722417563, 6.30978253779542156728101297956, 7.18888718297647721192763594914, 7.62196275802298861591436072358, 8.825846092924041988502122678303
