L(s) = 1 | + 4·11-s − 4·13-s − 12·23-s − 4·25-s + 4·37-s + 4·47-s − 2·49-s + 8·59-s − 4·61-s − 4·71-s − 16·73-s − 4·83-s − 8·97-s + 16·107-s − 12·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s − 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 1.10·13-s − 2.50·23-s − 4/5·25-s + 0.657·37-s + 0.583·47-s − 2/7·49-s + 1.04·59-s − 0.512·61-s − 0.474·71-s − 1.87·73-s − 0.439·83-s − 0.812·97-s + 1.54·107-s − 1.14·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.33·143-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.508890927228317107809155804325, −8.053531283266921403399232216356, −7.58362468779951906752252314034, −7.23581144856406587183158995420, −6.61804890147561498199371851340, −6.10217232780957668770129648496, −5.83252765072089165649169088141, −5.17530886323472951818277662326, −4.45114320120815774261463479937, −4.08546297328893572177165756463, −3.65956894261319686677630972148, −2.73025569532436935523506079390, −2.14349042559593556748670331493, −1.41065699043739306062040335688, 0,
1.41065699043739306062040335688, 2.14349042559593556748670331493, 2.73025569532436935523506079390, 3.65956894261319686677630972148, 4.08546297328893572177165756463, 4.45114320120815774261463479937, 5.17530886323472951818277662326, 5.83252765072089165649169088141, 6.10217232780957668770129648496, 6.61804890147561498199371851340, 7.23581144856406587183158995420, 7.58362468779951906752252314034, 8.053531283266921403399232216356, 8.508890927228317107809155804325