| L(s) = 1 | − 2-s + 7-s + 8-s + 3·9-s + 2·11-s − 14-s − 16-s + 7·17-s − 3·18-s − 2·22-s − 3·23-s + 12·29-s + 7·31-s − 7·34-s + 4·37-s − 14·41-s − 16·43-s + 3·46-s + 7·47-s − 6·49-s − 4·53-s + 56-s − 12·58-s + 14·59-s + 14·61-s − 7·62-s + 3·63-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.377·7-s + 0.353·8-s + 9-s + 0.603·11-s − 0.267·14-s − 1/4·16-s + 1.69·17-s − 0.707·18-s − 0.426·22-s − 0.625·23-s + 2.22·29-s + 1.25·31-s − 1.20·34-s + 0.657·37-s − 2.18·41-s − 2.43·43-s + 0.442·46-s + 1.02·47-s − 6/7·49-s − 0.549·53-s + 0.133·56-s − 1.57·58-s + 1.82·59-s + 1.79·61-s − 0.889·62-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330237931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330237931\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.97622288247141628084182420243, −11.29685536882335947917166577368, −10.45231566619500019162499392638, −10.24064832687969335178722925582, −9.829390911637106665841894047534, −9.742630520186105559131812883250, −8.786507306431124316737188132667, −8.441731629253052977476993093387, −8.047525054642078981280257566202, −7.68511200006167081488430338546, −6.80375593339708622047049576720, −6.72731984865423189383662011201, −6.04801085419458992481377128659, −5.08733248656995524803886834053, −4.92439453647826592655528096921, −4.11661633352941495922783940720, −3.55266354013516752286358624908, −2.75151428443267492863522121346, −1.62687141114993845125987658539, −1.05374713058931226300627199620,
1.05374713058931226300627199620, 1.62687141114993845125987658539, 2.75151428443267492863522121346, 3.55266354013516752286358624908, 4.11661633352941495922783940720, 4.92439453647826592655528096921, 5.08733248656995524803886834053, 6.04801085419458992481377128659, 6.72731984865423189383662011201, 6.80375593339708622047049576720, 7.68511200006167081488430338546, 8.047525054642078981280257566202, 8.441731629253052977476993093387, 8.786507306431124316737188132667, 9.742630520186105559131812883250, 9.829390911637106665841894047534, 10.24064832687969335178722925582, 10.45231566619500019162499392638, 11.29685536882335947917166577368, 11.97622288247141628084182420243
