| L(s) = 1 | + 4·5-s − 7-s − 4·11-s − 6·17-s − 4·19-s − 6·23-s + 11·25-s − 4·35-s + 6·37-s + 12·41-s + 4·43-s − 6·47-s + 49-s + 8·53-s − 16·55-s − 14·59-s − 14·61-s − 4·67-s − 12·71-s − 6·73-s + 4·77-s + 8·79-s − 6·83-s − 24·85-s + 16·89-s − 16·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 1.20·11-s − 1.45·17-s − 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.676·35-s + 0.986·37-s + 1.87·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 1.09·53-s − 2.15·55-s − 1.82·59-s − 1.79·61-s − 0.488·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 0.658·83-s − 2.60·85-s + 1.69·89-s − 1.64·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735139345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735139345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−13.78285940758175, −13.35257121533107, −13.14233534750179, −12.59927564378789, −12.16472600978183, −11.24891289715468, −10.67243365884662, −10.57157601741666, −9.919410763975069, −9.439877486924205, −9.023954147176296, −8.548889167207318, −7.765974013010443, −7.375268206808145, −6.403525767046846, −6.250879873474223, −5.831032089600887, −5.203294062519908, −4.475788375374662, −4.176828520372449, −2.930964621242341, −2.620626400109563, −2.077432899483368, −1.521322357050926, −0.3904913474755661,
0.3904913474755661, 1.521322357050926, 2.077432899483368, 2.620626400109563, 2.930964621242341, 4.176828520372449, 4.475788375374662, 5.203294062519908, 5.831032089600887, 6.250879873474223, 6.403525767046846, 7.375268206808145, 7.765974013010443, 8.548889167207318, 9.023954147176296, 9.439877486924205, 9.919410763975069, 10.57157601741666, 10.67243365884662, 11.24891289715468, 12.16472600978183, 12.59927564378789, 13.14233534750179, 13.35257121533107, 13.78285940758175
