| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 5·7-s + 4·8-s − 2·11-s + 2·12-s + 5·13-s + 10·14-s + 8·16-s + 2·17-s + 5·21-s − 4·22-s + 6·23-s + 4·24-s + 10·26-s − 27-s + 10·28-s + 4·29-s − 14·31-s + 8·32-s − 2·33-s + 4·34-s − 2·37-s + 5·39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.88·7-s + 1.41·8-s − 0.603·11-s + 0.577·12-s + 1.38·13-s + 2.67·14-s + 2·16-s + 0.485·17-s + 1.09·21-s − 0.852·22-s + 1.25·23-s + 0.816·24-s + 1.96·26-s − 0.192·27-s + 1.88·28-s + 0.742·29-s − 2.51·31-s + 1.41·32-s − 0.348·33-s + 0.685·34-s − 0.328·37-s + 0.800·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.742973562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.742973562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.43049998549944312341152511767, −10.06129316957753086885021350471, −9.107452310553927352648300173741, −8.909474866293837622326527584947, −8.317201609926387403092252443598, −8.314016588961537754189340428512, −7.43343234387023088586316125450, −7.34346679751807925245553439464, −7.10806293785033928768809006999, −6.00924555963716943950519564800, −5.76168188006061868435692921764, −5.28621929717501118482480550877, −5.01430051905080101940031568995, −4.37061448652907077986563926366, −4.15663118191157555751207607762, −3.52007807534941939497845543205, −3.06367625118072500175515010588, −2.35636734987625701627620210026, −1.43869885505388164874787171921, −1.43019433965512098771345422451,
1.43019433965512098771345422451, 1.43869885505388164874787171921, 2.35636734987625701627620210026, 3.06367625118072500175515010588, 3.52007807534941939497845543205, 4.15663118191157555751207607762, 4.37061448652907077986563926366, 5.01430051905080101940031568995, 5.28621929717501118482480550877, 5.76168188006061868435692921764, 6.00924555963716943950519564800, 7.10806293785033928768809006999, 7.34346679751807925245553439464, 7.43343234387023088586316125450, 8.314016588961537754189340428512, 8.317201609926387403092252443598, 8.909474866293837622326527584947, 9.107452310553927352648300173741, 10.06129316957753086885021350471, 10.43049998549944312341152511767
