L(s) = 1 | + 2·3-s + 2·4-s + 2·5-s + 2·7-s + 3·9-s − 2·11-s + 4·12-s + 4·13-s + 4·15-s − 6·17-s − 2·19-s + 4·20-s + 4·21-s + 6·23-s + 3·25-s + 4·27-s + 4·28-s − 6·29-s + 4·31-s − 4·33-s + 4·35-s + 6·36-s + 4·37-s + 8·39-s + 10·43-s − 4·44-s + 6·45-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 4-s + 0.894·5-s + 0.755·7-s + 9-s − 0.603·11-s + 1.15·12-s + 1.10·13-s + 1.03·15-s − 1.45·17-s − 0.458·19-s + 0.894·20-s + 0.872·21-s + 1.25·23-s + 3/5·25-s + 0.769·27-s + 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s + 36-s + 0.657·37-s + 1.28·39-s + 1.52·43-s − 0.603·44-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.096270198\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.096270198\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 121 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 147 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 120 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 189 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.909423917098188679026140748774, −9.588387219777469568733263832116, −8.953093057634432748978205323460, −8.826669711460802437776559137852, −8.511724097226487857608151422075, −7.937880742304264481796786981239, −7.58398871588602975228055763061, −7.04943385573894750509215927666, −6.82108169352892721128060972125, −6.22703458914111482835758646618, −5.95654102187777848031695487941, −5.36943391136335975818755921160, −4.57098493480731309504085897531, −4.56227699011763043193954203653, −3.74553808317078905728907180294, −3.09112922898959823255317542420, −2.66938400307806483527995700217, −2.06597829935012067119608461118, −1.89371464819143966480518331598, −1.04009537592592898962232236952,
1.04009537592592898962232236952, 1.89371464819143966480518331598, 2.06597829935012067119608461118, 2.66938400307806483527995700217, 3.09112922898959823255317542420, 3.74553808317078905728907180294, 4.56227699011763043193954203653, 4.57098493480731309504085897531, 5.36943391136335975818755921160, 5.95654102187777848031695487941, 6.22703458914111482835758646618, 6.82108169352892721128060972125, 7.04943385573894750509215927666, 7.58398871588602975228055763061, 7.937880742304264481796786981239, 8.511724097226487857608151422075, 8.826669711460802437776559137852, 8.953093057634432748978205323460, 9.588387219777469568733263832116, 9.909423917098188679026140748774