L(s) = 1 | + 3-s + 7-s − 2·9-s + 7·13-s − 5·19-s + 21-s + 25-s − 5·27-s + 8·31-s − 37-s + 7·39-s + 2·43-s − 7·49-s − 5·57-s + 61-s − 2·63-s + 14·67-s − 7·73-s + 75-s + 5·79-s + 81-s + 7·91-s + 8·93-s + 6·97-s − 7·103-s − 5·109-s − 111-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.94·13-s − 1.14·19-s + 0.218·21-s + 1/5·25-s − 0.962·27-s + 1.43·31-s − 0.164·37-s + 1.12·39-s + 0.304·43-s − 49-s − 0.662·57-s + 0.128·61-s − 0.251·63-s + 1.71·67-s − 0.819·73-s + 0.115·75-s + 0.562·79-s + 1/9·81-s + 0.733·91-s + 0.829·93-s + 0.609·97-s − 0.689·103-s − 0.478·109-s − 0.0949·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.658927385\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.658927385\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 7 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.099019925669871420899449770436, −7.951212596030188259642977683744, −7.15814873220015976581596522215, −6.64824623564342855796275616133, −6.31010011945207620075425791788, −5.92629860007276033402748165187, −5.45363719479375583298028614978, −4.85785522232237836938717516607, −4.29938770688531445955014253560, −3.89112949214453868107542740841, −3.33145212151059683709197545430, −2.88422434592188062474744063642, −2.18571663082976232428626251441, −1.60889689728010078756885159562, −0.72866282716440172154646166569,
0.72866282716440172154646166569, 1.60889689728010078756885159562, 2.18571663082976232428626251441, 2.88422434592188062474744063642, 3.33145212151059683709197545430, 3.89112949214453868107542740841, 4.29938770688531445955014253560, 4.85785522232237836938717516607, 5.45363719479375583298028614978, 5.92629860007276033402748165187, 6.31010011945207620075425791788, 6.64824623564342855796275616133, 7.15814873220015976581596522215, 7.951212596030188259642977683744, 8.099019925669871420899449770436