| L(s) = 1 | − 4·4-s + 7-s − 5·9-s − 6·11-s + 12·16-s − 12·23-s + 25-s − 4·28-s + 6·29-s + 20·36-s + 4·37-s − 20·43-s + 24·44-s + 49-s + 24·53-s − 5·63-s − 32·64-s − 8·67-s − 6·77-s − 2·79-s + 16·81-s + 48·92-s + 30·99-s − 4·100-s + 12·107-s − 14·109-s + 12·112-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.377·7-s − 5/3·9-s − 1.80·11-s + 3·16-s − 2.50·23-s + 1/5·25-s − 0.755·28-s + 1.11·29-s + 10/3·36-s + 0.657·37-s − 3.04·43-s + 3.61·44-s + 1/7·49-s + 3.29·53-s − 0.629·63-s − 4·64-s − 0.977·67-s − 0.683·77-s − 0.225·79-s + 16/9·81-s + 5.00·92-s + 3.01·99-s − 2/5·100-s + 1.16·107-s − 1.34·109-s + 1.13·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−11.72263911909931664995468447198, −10.46766423688804548750268495569, −10.26175613408033680908306725927, −9.833804192038460581265801340645, −8.722193961422815471879095064652, −8.679498853252053444462430241617, −7.980947296320422939728903060714, −7.80156257853413862136981277007, −6.30078072165756181986797640838, −5.47907564040983295563221596641, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −3.61689003045514298236484607491, −2.61198254943438177491615875407, 0,
2.61198254943438177491615875407, 3.61689003045514298236484607491, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 5.47907564040983295563221596641, 6.30078072165756181986797640838, 7.80156257853413862136981277007, 7.980947296320422939728903060714, 8.679498853252053444462430241617, 8.722193961422815471879095064652, 9.833804192038460581265801340645, 10.26175613408033680908306725927, 10.46766423688804548750268495569, 11.72263911909931664995468447198
