| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·17-s + 6·19-s + 2·21-s + 6·23-s + 27-s + 4·29-s + 8·31-s − 6·37-s − 6·41-s + 4·43-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 6·57-s − 2·61-s + 2·63-s + 4·67-s + 6·69-s − 8·71-s − 16·79-s + 81-s + 4·83-s + 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.970·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.794·57-s − 0.256·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s − 1.80·79-s + 1/9·81-s + 0.439·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.042154335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.042154335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.17019131235612, −12.56019925443132, −12.03957651872658, −11.70174133649414, −11.28279649124292, −10.63490227296293, −10.11808089656180, −9.850410024006262, −9.218985687240331, −8.751805258850048, −8.261501683892336, −7.869661162167494, −7.410372573138330, −6.894235430776494, −6.395726636540060, −5.673821057646595, −5.062232944291214, −4.871015558475023, −4.211233221965633, −3.389753520336723, −3.107773798382557, −2.607427480174917, −1.645095396894785, −1.315207677091654, −0.6419727143190675,
0.6419727143190675, 1.315207677091654, 1.645095396894785, 2.607427480174917, 3.107773798382557, 3.389753520336723, 4.211233221965633, 4.871015558475023, 5.062232944291214, 5.673821057646595, 6.395726636540060, 6.894235430776494, 7.410372573138330, 7.869661162167494, 8.261501683892336, 8.751805258850048, 9.218985687240331, 9.850410024006262, 10.11808089656180, 10.63490227296293, 11.28279649124292, 11.70174133649414, 12.03957651872658, 12.56019925443132, 13.17019131235612
