| L(s) = 1 | + 3·3-s − 7-s + 6·9-s + 2·19-s − 3·21-s − 6·25-s + 9·27-s + 9·29-s + 12·31-s + 4·37-s − 13·47-s − 6·49-s + 7·53-s + 6·57-s + 12·59-s − 6·63-s − 18·75-s + 9·81-s + 7·83-s + 27·87-s + 36·93-s − 4·103-s + 6·109-s + 12·111-s − 23·113-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s + 0.458·19-s − 0.654·21-s − 6/5·25-s + 1.73·27-s + 1.67·29-s + 2.15·31-s + 0.657·37-s − 1.89·47-s − 6/7·49-s + 0.961·53-s + 0.794·57-s + 1.56·59-s − 0.755·63-s − 2.07·75-s + 81-s + 0.768·83-s + 2.89·87-s + 3.73·93-s − 0.394·103-s + 0.574·109-s + 1.13·111-s − 2.16·113-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.547256621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.547256621\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 124 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
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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
−8.548230067961645956705501449877, −8.306010837411391199216828365596, −8.032906948223741637121915783509, −7.54465133384952977356413543268, −6.95138468952159052229297452621, −6.47633324804228939066385032642, −6.16688188734518866815995072515, −5.23947737914702759713385737212, −4.75653803692371289558715364840, −4.12285424342029162394495691017, −3.72073370280889662561278619323, −2.93135528362509942057264608008, −2.74841568003733760151668622793, −1.95696021900071073763756290239, −1.04674296635555244402881413397,
1.04674296635555244402881413397, 1.95696021900071073763756290239, 2.74841568003733760151668622793, 2.93135528362509942057264608008, 3.72073370280889662561278619323, 4.12285424342029162394495691017, 4.75653803692371289558715364840, 5.23947737914702759713385737212, 6.16688188734518866815995072515, 6.47633324804228939066385032642, 6.95138468952159052229297452621, 7.54465133384952977356413543268, 8.032906948223741637121915783509, 8.306010837411391199216828365596, 8.548230067961645956705501449877
