L(s) = 1 | − 7-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 6·29-s − 8·31-s + 10·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s + 8·59-s − 14·61-s + 3·63-s − 12·67-s + 16·71-s − 2·73-s + 4·77-s + 8·79-s + 9·81-s + 8·83-s + 10·89-s − 6·91-s − 2·97-s + 12·99-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.04·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 81-s + 0.878·83-s + 1.05·89-s − 0.628·91-s − 0.203·97-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.401898985\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.401898985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.804167587109299746375991715623, −8.107288137075358285056657641589, −7.41297246540534830906507852807, −6.24190347389181960956817365315, −5.91578381006435814657965447568, −4.99516684089165156540912778411, −3.94248355024955497277318994877, −3.07578209230613039021513049561, −2.26639998875518159229729259543, −0.71428970411115330695948810032,
0.71428970411115330695948810032, 2.26639998875518159229729259543, 3.07578209230613039021513049561, 3.94248355024955497277318994877, 4.99516684089165156540912778411, 5.91578381006435814657965447568, 6.24190347389181960956817365315, 7.41297246540534830906507852807, 8.107288137075358285056657641589, 8.804167587109299746375991715623