| L(s) = 1 | − 5-s + 3·7-s − 2·11-s + 13-s + 3·17-s + 2·19-s − 4·23-s − 4·25-s − 2·29-s + 4·31-s − 3·35-s + 5·37-s + 12·41-s + 7·43-s + 9·47-s + 2·49-s − 4·53-s + 2·55-s − 6·59-s − 4·61-s − 65-s − 10·67-s + 15·71-s − 2·73-s − 6·77-s − 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s + 0.718·31-s − 0.507·35-s + 0.821·37-s + 1.87·41-s + 1.06·43-s + 1.31·47-s + 2/7·49-s − 0.549·53-s + 0.269·55-s − 0.781·59-s − 0.512·61-s − 0.124·65-s − 1.22·67-s + 1.78·71-s − 0.234·73-s − 0.683·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.960236682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960236682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.268472913194180703129425063211, −7.74455014790952466693900217268, −7.43844121148243357403266871461, −6.06419379069511118441962565825, −5.60703236762079894903659847623, −4.59882498172775545774387253041, −4.05033961651429934645679739631, −2.95935495755402668926350585672, −1.97342449528249493763599188279, −0.835497600438090992100165890813,
0.835497600438090992100165890813, 1.97342449528249493763599188279, 2.95935495755402668926350585672, 4.05033961651429934645679739631, 4.59882498172775545774387253041, 5.60703236762079894903659847623, 6.06419379069511118441962565825, 7.43844121148243357403266871461, 7.74455014790952466693900217268, 8.268472913194180703129425063211
