| L(s) = 1 | + 6·3-s − 6·5-s − 14·7-s + 27·9-s − 6·11-s − 16·13-s − 36·15-s − 6·17-s + 64·19-s − 84·21-s − 6·23-s − 166·25-s + 108·27-s + 252·29-s − 40·31-s − 36·33-s + 84·35-s + 248·37-s − 96·39-s − 450·41-s + 376·43-s − 162·45-s + 12·47-s + 147·49-s − 36·51-s + 1.10e3·53-s + 36·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.536·5-s − 0.755·7-s + 9-s − 0.164·11-s − 0.341·13-s − 0.619·15-s − 0.0856·17-s + 0.772·19-s − 0.872·21-s − 0.0543·23-s − 1.32·25-s + 0.769·27-s + 1.61·29-s − 0.231·31-s − 0.189·33-s + 0.405·35-s + 1.10·37-s − 0.394·39-s − 1.71·41-s + 1.33·43-s − 0.536·45-s + 0.0372·47-s + 3/7·49-s − 0.0988·51-s + 2.86·53-s + 0.0882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.083114591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.083114591\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 202 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 7870 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T - 13890 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 248 T + 98214 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 141790 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1104 T + 602230 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 428 T + 425886 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 954 T + 13106 p T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 572 T + 901662 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.400492301415239746290306324714, −9.006743623146583624947976169006, −8.585709901700530857523540634968, −8.327276651260711185119573687609, −7.69918104545993394763569895851, −7.64373570667051806031404691358, −6.94642260521056048716137724396, −6.88863218169885432462384766345, −6.11866534038416535760956524620, −5.82981123944744025530687786941, −5.09162289729310630041079672206, −4.80942102053493307172613486536, −4.03738302754361848740736742301, −3.81037685308484063842822231186, −3.39639940542760708770362563503, −2.82214717362446157306408820133, −2.29488723284989378883591027340, −1.99491353710157120667241029432, −0.73234532025868511622353979724, −0.71519488616983047452452648080,
0.71519488616983047452452648080, 0.73234532025868511622353979724, 1.99491353710157120667241029432, 2.29488723284989378883591027340, 2.82214717362446157306408820133, 3.39639940542760708770362563503, 3.81037685308484063842822231186, 4.03738302754361848740736742301, 4.80942102053493307172613486536, 5.09162289729310630041079672206, 5.82981123944744025530687786941, 6.11866534038416535760956524620, 6.88863218169885432462384766345, 6.94642260521056048716137724396, 7.64373570667051806031404691358, 7.69918104545993394763569895851, 8.327276651260711185119573687609, 8.585709901700530857523540634968, 9.006743623146583624947976169006, 9.400492301415239746290306324714
