| L(s) = 1 | + 2·5-s − 4·11-s − 4·23-s + 3·25-s + 4·29-s − 8·31-s + 12·37-s + 12·41-s − 4·47-s − 4·53-s − 8·55-s + 8·67-s − 4·71-s + 8·73-s + 12·79-s − 4·83-s + 4·89-s + 24·97-s + 12·101-s + 16·103-s − 28·107-s + 12·109-s + 4·113-s − 8·115-s − 2·121-s + 4·125-s + 127-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.834·23-s + 3/5·25-s + 0.742·29-s − 1.43·31-s + 1.97·37-s + 1.87·41-s − 0.583·47-s − 0.549·53-s − 1.07·55-s + 0.977·67-s − 0.474·71-s + 0.936·73-s + 1.35·79-s − 0.439·83-s + 0.423·89-s + 2.43·97-s + 1.19·101-s + 1.57·103-s − 2.70·107-s + 1.14·109-s + 0.376·113-s − 0.746·115-s − 0.181·121-s + 0.357·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.829870407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.829870407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 330 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.004377945655238211497940725209, −7.64502600041065327525926039793, −7.16428960886426726981757219097, −7.05543708646167001421184157084, −6.30348549942847989682143095830, −6.26663794823812533110758647123, −5.75483452816342466203704154143, −5.74863360965845517449927925351, −5.07859649846978330899244069306, −4.92485246716499092246916359288, −4.43493108589399276299310814215, −4.19352591588397598304322694814, −3.49046086790282830669119849348, −3.33572374009590112191671505597, −2.59655169838587567238973954623, −2.55360874319574127427233037320, −1.92605393344512545847480364698, −1.75503437640563258844114717520, −0.75941724888655398787153522553, −0.59633636640456954668433124136,
0.59633636640456954668433124136, 0.75941724888655398787153522553, 1.75503437640563258844114717520, 1.92605393344512545847480364698, 2.55360874319574127427233037320, 2.59655169838587567238973954623, 3.33572374009590112191671505597, 3.49046086790282830669119849348, 4.19352591588397598304322694814, 4.43493108589399276299310814215, 4.92485246716499092246916359288, 5.07859649846978330899244069306, 5.74863360965845517449927925351, 5.75483452816342466203704154143, 6.26663794823812533110758647123, 6.30348549942847989682143095830, 7.05543708646167001421184157084, 7.16428960886426726981757219097, 7.64502600041065327525926039793, 8.004377945655238211497940725209
