L(s) = 1 | + 6·7-s + 4·13-s − 8·19-s + 2·25-s + 14·31-s + 4·43-s + 14·49-s − 12·67-s + 4·73-s + 22·79-s + 24·91-s + 4·97-s + 2·103-s + 28·109-s − 10·121-s + 127-s + 131-s − 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
L(s) = 1 | + 2.26·7-s + 1.10·13-s − 1.83·19-s + 2/5·25-s + 2.51·31-s + 0.609·43-s + 2·49-s − 1.46·67-s + 0.468·73-s + 2.47·79-s + 2.51·91-s + 0.406·97-s + 0.197·103-s + 2.68·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.272969326\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.272969326\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.008048260768210052939274358514, −7.79665728268307322575478399455, −7.16169120733400146658632687732, −6.55357030574168677103143173928, −6.21915628298422877722468036549, −5.91363507099060333975893632266, −5.07194235330730713596397747797, −4.85966830487633530414981933212, −4.43093511781586946568505278083, −4.06130048053471263314885142781, −3.39443502494172850844054158242, −2.58024364216897450967331891613, −2.10214891919535945429491099668, −1.48199850184422005196283542697, −0.871151168746476274929101842240,
0.871151168746476274929101842240, 1.48199850184422005196283542697, 2.10214891919535945429491099668, 2.58024364216897450967331891613, 3.39443502494172850844054158242, 4.06130048053471263314885142781, 4.43093511781586946568505278083, 4.85966830487633530414981933212, 5.07194235330730713596397747797, 5.91363507099060333975893632266, 6.21915628298422877722468036549, 6.55357030574168677103143173928, 7.16169120733400146658632687732, 7.79665728268307322575478399455, 8.008048260768210052939274358514