| L(s) = 1 | + 4-s + 4·7-s + 6·11-s + 16-s + 2·25-s + 4·28-s + 2·37-s − 24·41-s + 6·44-s + 6·47-s + 7·49-s + 64-s − 2·67-s + 22·73-s + 24·77-s + 6·83-s + 2·100-s − 12·101-s + 18·107-s + 4·112-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s + 1.80·11-s + 1/4·16-s + 2/5·25-s + 0.755·28-s + 0.328·37-s − 3.74·41-s + 0.904·44-s + 0.875·47-s + 49-s + 1/8·64-s − 0.244·67-s + 2.57·73-s + 2.73·77-s + 0.658·83-s + 1/5·100-s − 1.19·101-s + 1.74·107-s + 0.377·112-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.085158981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.085158981\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−8.517368134088958397464040397578, −8.142736804886147617607793386324, −7.78887991034520505444722328811, −7.10243765701138556180485490357, −6.68337315430167712818312824576, −6.53146858043024313108073230737, −5.75341972445244853247371849909, −5.21090285825918176155570782179, −4.83012210430459997060037036510, −4.23785601187187265518884388715, −3.68209827831231977771933611840, −3.18340095371178264301298210158, −2.16309732307087400377937762817, −1.70193189217264372363787897976, −1.06492565906145681641297652547,
1.06492565906145681641297652547, 1.70193189217264372363787897976, 2.16309732307087400377937762817, 3.18340095371178264301298210158, 3.68209827831231977771933611840, 4.23785601187187265518884388715, 4.83012210430459997060037036510, 5.21090285825918176155570782179, 5.75341972445244853247371849909, 6.53146858043024313108073230737, 6.68337315430167712818312824576, 7.10243765701138556180485490357, 7.78887991034520505444722328811, 8.142736804886147617607793386324, 8.517368134088958397464040397578
