| L(s) = 1 | − 4·4-s + 20·5-s − 44·11-s + 16·16-s + 162·19-s − 80·20-s + 275·25-s − 540·29-s − 226·31-s − 812·41-s + 176·44-s + 490·49-s − 880·55-s − 548·59-s + 82·61-s − 64·64-s + 780·71-s − 648·76-s + 2.43e3·79-s + 320·80-s + 1.02e3·89-s + 3.24e3·95-s − 1.10e3·100-s + 1.18e3·101-s + 1.87e3·109-s + 2.16e3·116-s − 1.21e3·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1.20·11-s + 1/4·16-s + 1.95·19-s − 0.894·20-s + 11/5·25-s − 3.45·29-s − 1.30·31-s − 3.09·41-s + 0.603·44-s + 10/7·49-s − 2.15·55-s − 1.20·59-s + 0.172·61-s − 1/8·64-s + 1.30·71-s − 0.978·76-s + 3.46·79-s + 0.447·80-s + 1.22·89-s + 3.49·95-s − 1.09·100-s + 1.16·101-s + 1.64·109-s + 1.72·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.413052899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.413052899\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3494 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9777 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 81 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1533 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 113 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 93562 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 406 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204510 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 277873 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 274 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 41 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 493942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 390 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 386158 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1215 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 888549 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 514 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )( 1 + 594 T + p^{3} T^{2} ) \) |
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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
−11.88507308355693156150265111714, −10.89701603894004588911498375978, −10.82531742626522411780781710984, −10.27454050869939613865480052217, −9.595561380693217810864427784913, −9.388816557340488679621987365299, −9.229731571256387630677279966072, −8.361194709096518156920100066945, −7.84712748363474887587079452012, −7.20670672256224985504876525365, −6.92467697947401491138844987242, −5.77264230111407691600485721134, −5.73970337664977508552875413821, −5.18264746757200096646376804967, −4.84960529878972625627112370948, −3.44207948087384888844778966852, −3.37103625578538936376132351742, −2.02686855489427722323571907809, −1.87642173563245860458397399743, −0.58975226664755717721443566585,
0.58975226664755717721443566585, 1.87642173563245860458397399743, 2.02686855489427722323571907809, 3.37103625578538936376132351742, 3.44207948087384888844778966852, 4.84960529878972625627112370948, 5.18264746757200096646376804967, 5.73970337664977508552875413821, 5.77264230111407691600485721134, 6.92467697947401491138844987242, 7.20670672256224985504876525365, 7.84712748363474887587079452012, 8.361194709096518156920100066945, 9.229731571256387630677279966072, 9.388816557340488679621987365299, 9.595561380693217810864427784913, 10.27454050869939613865480052217, 10.82531742626522411780781710984, 10.89701603894004588911498375978, 11.88507308355693156150265111714
