| L(s) = 1 | − 2·4-s − 5·9-s − 8·13-s + 4·16-s + 12·17-s − 10·25-s − 12·29-s + 10·36-s + 2·37-s − 18·41-s − 13·49-s + 16·52-s − 6·53-s + 16·61-s − 8·64-s − 24·68-s + 22·73-s + 16·81-s + 12·89-s + 16·97-s + 20·100-s + 6·101-s + 4·109-s − 12·113-s + 24·116-s + 40·117-s − 13·121-s + ⋯ |
| L(s) = 1 | − 4-s − 5/3·9-s − 2.21·13-s + 16-s + 2.91·17-s − 2·25-s − 2.22·29-s + 5/3·36-s + 0.328·37-s − 2.81·41-s − 1.85·49-s + 2.21·52-s − 0.824·53-s + 2.04·61-s − 64-s − 2.91·68-s + 2.57·73-s + 16/9·81-s + 1.27·89-s + 1.62·97-s + 2·100-s + 0.597·101-s + 0.383·109-s − 1.12·113-s + 2.22·116-s + 3.69·117-s − 1.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.19326127043497776672436763272, −9.819966817497618882768827794755, −9.666580450452829708455786810978, −9.032345851694468357832029856913, −8.152382970199818474974946940954, −7.83252666101026187146437096539, −7.59911177067371416247669368567, −6.49509806030834552919855622210, −5.50302825732666851289149768404, −5.44973416215471530225317007593, −4.91803715229452696224797784043, −3.50910294340479626549928321873, −3.42634561262787911027610453075, −2.09291945273234411794912146252, 0,
2.09291945273234411794912146252, 3.42634561262787911027610453075, 3.50910294340479626549928321873, 4.91803715229452696224797784043, 5.44973416215471530225317007593, 5.50302825732666851289149768404, 6.49509806030834552919855622210, 7.59911177067371416247669368567, 7.83252666101026187146437096539, 8.152382970199818474974946940954, 9.032345851694468357832029856913, 9.666580450452829708455786810978, 9.819966817497618882768827794755, 10.19326127043497776672436763272
