| L(s) = 1 | − 5·5-s − 36·7-s + 64·11-s + 65·13-s − 118·17-s − 56·19-s + 160·23-s + 125·25-s − 57·29-s − 164·31-s + 180·35-s − 642·37-s − 246·41-s + 8·43-s + 84·47-s + 343·49-s − 956·53-s − 320·55-s − 32·59-s − 415·61-s − 325·65-s + 220·67-s − 1.76e3·71-s − 154·73-s − 2.30e3·77-s + 80·79-s + 1.26e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.94·7-s + 1.75·11-s + 1.38·13-s − 1.68·17-s − 0.676·19-s + 1.45·23-s + 25-s − 0.364·29-s − 0.950·31-s + 0.869·35-s − 2.85·37-s − 0.937·41-s + 0.0283·43-s + 0.260·47-s + 49-s − 2.47·53-s − 0.784·55-s − 0.0706·59-s − 0.871·61-s − 0.620·65-s + 0.401·67-s − 2.95·71-s − 0.246·73-s − 3.40·77-s + 0.113·79-s + 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + p T - 4 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 36 T + 953 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 64 T + 2765 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 p T + p^{3} T^{2} )( 1 + 2 p T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 59 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 160 T + 13433 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 57 T - 21140 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 164 T - 2895 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 321 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 p T - 5 p^{2} T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 8 T - 79443 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 84 T - 96767 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 478 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 32 T - 204355 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 415 T - 54756 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 220 T - 252363 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 77 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 80 T - 486639 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1268 T + 1036037 T^{2} - 1268 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 123 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1346 T + 899043 T^{2} + 1346 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.839169454095030821309918712467, −9.265409057597001273735952598767, −9.011270049080328961142120369451, −8.910537215250439417566522291485, −8.456475314254387248150146007828, −7.69118229658529754219908224780, −6.83706932770647032749218830637, −6.80867921873977312441837034237, −6.54208165199055914159988448568, −6.17395448509503841003034052959, −5.38630470278168185298589111680, −4.81599373294753303160575422413, −3.96063624511826077621135810952, −3.93409683730492408081709713218, −3.08735014385458095505775890611, −3.05896849253703633554538680754, −1.70344418540752897933857746287, −1.34964812793529791332999279598, 0, 0,
1.34964812793529791332999279598, 1.70344418540752897933857746287, 3.05896849253703633554538680754, 3.08735014385458095505775890611, 3.93409683730492408081709713218, 3.96063624511826077621135810952, 4.81599373294753303160575422413, 5.38630470278168185298589111680, 6.17395448509503841003034052959, 6.54208165199055914159988448568, 6.80867921873977312441837034237, 6.83706932770647032749218830637, 7.69118229658529754219908224780, 8.456475314254387248150146007828, 8.910537215250439417566522291485, 9.011270049080328961142120369451, 9.265409057597001273735952598767, 9.839169454095030821309918712467
