| L(s) = 1 | + 3-s + 3·5-s − 4·7-s + 3·9-s − 3·11-s − 4·13-s + 3·15-s − 3·17-s − 19-s − 4·21-s − 3·23-s + 5·25-s + 8·27-s + 12·29-s + 7·31-s − 3·33-s − 12·35-s − 37-s − 4·39-s + 12·41-s + 8·43-s + 9·45-s + 9·47-s + 9·49-s − 3·51-s + 3·53-s − 9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 1.51·7-s + 9-s − 0.904·11-s − 1.10·13-s + 0.774·15-s − 0.727·17-s − 0.229·19-s − 0.872·21-s − 0.625·23-s + 25-s + 1.53·27-s + 2.22·29-s + 1.25·31-s − 0.522·33-s − 2.02·35-s − 0.164·37-s − 0.640·39-s + 1.87·41-s + 1.21·43-s + 1.34·45-s + 1.31·47-s + 9/7·49-s − 0.420·51-s + 0.412·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.149306983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149306983\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.11029907321283449217431173580, −10.60216375342673559217309081292, −10.19387669536850782414458407309, −9.916742350002390261269684926635, −9.750384490253818090029257775479, −9.213615508919535885049459952993, −8.511452802147931166586230345484, −8.473733715006804911329519167331, −7.42839370953603325488625656554, −7.21244568690470416737831698831, −6.66863239806856136155009035563, −6.14366164945005922025632076894, −5.87222181691419343099747319456, −5.06494523167841989402454105010, −4.46678606196403938125379725520, −4.10174630475570383932116435597, −2.86797232150328807388802996516, −2.69291512271974295727268605821, −2.25511409019656162972402365717, −0.907207386151134318322245084356,
0.907207386151134318322245084356, 2.25511409019656162972402365717, 2.69291512271974295727268605821, 2.86797232150328807388802996516, 4.10174630475570383932116435597, 4.46678606196403938125379725520, 5.06494523167841989402454105010, 5.87222181691419343099747319456, 6.14366164945005922025632076894, 6.66863239806856136155009035563, 7.21244568690470416737831698831, 7.42839370953603325488625656554, 8.473733715006804911329519167331, 8.511452802147931166586230345484, 9.213615508919535885049459952993, 9.750384490253818090029257775479, 9.916742350002390261269684926635, 10.19387669536850782414458407309, 10.60216375342673559217309081292, 11.11029907321283449217431173580
