| L(s) = 1 | − 16·4-s + 386·9-s − 672·11-s + 256·16-s − 940·19-s − 9.73e3·29-s − 1.47e4·31-s − 6.17e3·36-s + 1.24e4·41-s + 1.07e4·44-s − 2.40e3·49-s − 6.86e4·59-s + 4.89e4·61-s − 4.09e3·64-s + 5.64e4·71-s + 1.50e4·76-s − 8.57e4·79-s + 8.99e4·81-s − 5.34e4·89-s − 2.59e5·99-s + 1.98e5·101-s + 1.12e5·109-s + 1.55e5·116-s + 1.65e4·121-s + 2.35e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.58·9-s − 1.67·11-s + 1/4·16-s − 0.597·19-s − 2.14·29-s − 2.75·31-s − 0.794·36-s + 1.15·41-s + 0.837·44-s − 1/7·49-s − 2.56·59-s + 1.68·61-s − 1/8·64-s + 1.32·71-s + 0.298·76-s − 1.54·79-s + 1.52·81-s − 0.715·89-s − 2.65·99-s + 1.93·101-s + 0.908·109-s + 1.07·116-s + 0.102·121-s + 1.37·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1549436771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1549436771\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 386 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 336 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 401530 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 713950 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 470 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4767314 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4866 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7372 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 66660986 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6222 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 280297270 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 455406670 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 550575578 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 34302 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 24476 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2395677910 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28224 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4133168782 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 42872 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6638900482 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 26730 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16886428030 T^{2} + p^{10} T^{4} \) |
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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
−10.91482675832083380433318711608, −10.46145592128473862379116065917, −9.942312286694969332811029044726, −9.507919612076122144680995630556, −9.156521651098255685372791119377, −8.637798875245618110953691044252, −7.86494831696282535382081494915, −7.58230792219091195523749870964, −7.30815962794208799690013151706, −6.71239310414796954393900770215, −5.79951012264723712169761647065, −5.58611539466392351393877406554, −4.92062754587351326930877978929, −4.45528708118665863259116259482, −3.78753495234827291574518872787, −3.43845076254983193078730196317, −2.36075563566662616340105700767, −1.92521932175293161940504432964, −1.20182873494173093676577814337, −0.10424926927285969967064912207,
0.10424926927285969967064912207, 1.20182873494173093676577814337, 1.92521932175293161940504432964, 2.36075563566662616340105700767, 3.43845076254983193078730196317, 3.78753495234827291574518872787, 4.45528708118665863259116259482, 4.92062754587351326930877978929, 5.58611539466392351393877406554, 5.79951012264723712169761647065, 6.71239310414796954393900770215, 7.30815962794208799690013151706, 7.58230792219091195523749870964, 7.86494831696282535382081494915, 8.637798875245618110953691044252, 9.156521651098255685372791119377, 9.507919612076122144680995630556, 9.942312286694969332811029044726, 10.46145592128473862379116065917, 10.91482675832083380433318711608
