| L(s) = 1 | − 4·2-s − 18·3-s − 20·4-s + 50·5-s + 72·6-s − 98·7-s + 96·8-s + 243·9-s − 200·10-s − 176·11-s + 360·12-s + 692·13-s + 392·14-s − 900·15-s − 176·16-s − 428·17-s − 972·18-s − 1.82e3·19-s − 1.00e3·20-s + 1.76e3·21-s + 704·22-s + 8.03e3·23-s − 1.72e3·24-s + 1.87e3·25-s − 2.76e3·26-s − 2.91e3·27-s + 1.96e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 5/8·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s + 0.530·8-s + 9-s − 0.632·10-s − 0.438·11-s + 0.721·12-s + 1.13·13-s + 0.534·14-s − 1.03·15-s − 0.171·16-s − 0.359·17-s − 0.707·18-s − 1.15·19-s − 0.559·20-s + 0.872·21-s + 0.310·22-s + 3.16·23-s − 0.612·24-s + 3/5·25-s − 0.803·26-s − 0.769·27-s + 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.028540874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028540874\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{2} T + 9 p^{2} T^{2} + p^{7} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 16 p T + 9846 T^{2} + 16 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 692 T + 856894 T^{2} - 692 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 428 T + 1236582 T^{2} + 428 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 96 p T + 5676294 T^{2} + 96 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8032 T + 1215330 p T^{2} - 8032 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2948 T + 32688446 T^{2} + 2948 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14760 T + 109231790 T^{2} - 14760 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 156 T - 37875634 T^{2} - 156 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5980 T + 145983702 T^{2} + 5980 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 25672 T + 418423654 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21904 T + 571171070 T^{2} - 21904 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 53948 T + 1562783310 T^{2} + 53948 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 26296 T + 547660454 T^{2} - 26296 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16788 T + 1437884126 T^{2} + 16788 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16264 T + 2696002390 T^{2} - 16264 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22264 T + 3731796926 T^{2} - 22264 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 28684 T + 3989062102 T^{2} - 28684 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 64368 T + 3610665822 T^{2} + 64368 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 75848 T + 5983113110 T^{2} - 75848 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 32964 T + 2468206070 T^{2} - 32964 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 242604 T + 31399386246 T^{2} - 242604 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.93000311614125282560999347193, −12.81917254923099625769222501251, −12.06328763179390308742422516348, −11.18844784618042294364199713848, −10.79962581755129958134827989774, −10.59036539813377050944189016428, −9.627402990841955300321633756068, −9.557251115582868590979026178056, −8.678618622317326735134350706938, −8.586540656350304798626810748094, −7.43105848256793532849026129018, −6.73568794705894968021060305638, −6.23107272331740743616176826720, −5.84324531477534584832564795577, −4.76294761830137472458184808671, −4.62764199857520139583130003735, −3.33951049932083851179196123594, −2.38401151159402294079114308652, −0.991682293632231292704255564167, −0.63774534019035493078154723443,
0.63774534019035493078154723443, 0.991682293632231292704255564167, 2.38401151159402294079114308652, 3.33951049932083851179196123594, 4.62764199857520139583130003735, 4.76294761830137472458184808671, 5.84324531477534584832564795577, 6.23107272331740743616176826720, 6.73568794705894968021060305638, 7.43105848256793532849026129018, 8.586540656350304798626810748094, 8.678618622317326735134350706938, 9.557251115582868590979026178056, 9.627402990841955300321633756068, 10.59036539813377050944189016428, 10.79962581755129958134827989774, 11.18844784618042294364199713848, 12.06328763179390308742422516348, 12.81917254923099625769222501251, 12.93000311614125282560999347193
