| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s + 8·19-s − 4·29-s + 8·31-s + 36-s + 4·41-s + 8·44-s − 2·49-s − 24·59-s + 4·61-s − 64-s − 8·71-s − 8·76-s + 24·79-s + 81-s − 20·89-s + 8·99-s − 20·101-s − 20·109-s + 4·116-s + 26·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 1.83·19-s − 0.742·29-s + 1.43·31-s + 1/6·36-s + 0.624·41-s + 1.20·44-s − 2/7·49-s − 3.12·59-s + 0.512·61-s − 1/8·64-s − 0.949·71-s − 0.917·76-s + 2.70·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s − 1.99·101-s − 1.91·109-s + 0.371·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9237080270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9237080270\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.484350459542729175077605349481, −8.526696157553920822526586781070, −8.181018576289516401880431443991, −8.061682512454208525152312623585, −7.69102384465031280252182372834, −7.15500795569719800791717256822, −7.08869841627912843901530571541, −6.12268663313949988592973792494, −6.03206777548489003367142518978, −5.41144885246067512606594430658, −5.28032134863114357356395140543, −4.78564837535476125633908619600, −4.50783288174807522656531790605, −3.84573451300330212044988773821, −3.24443869805087509369657731728, −2.85764367860221137790042945007, −2.67367673038326662384389447723, −1.87432373695221615464050964211, −1.14234966438259630109544124512, −0.34941243309231677094097767329,
0.34941243309231677094097767329, 1.14234966438259630109544124512, 1.87432373695221615464050964211, 2.67367673038326662384389447723, 2.85764367860221137790042945007, 3.24443869805087509369657731728, 3.84573451300330212044988773821, 4.50783288174807522656531790605, 4.78564837535476125633908619600, 5.28032134863114357356395140543, 5.41144885246067512606594430658, 6.03206777548489003367142518978, 6.12268663313949988592973792494, 7.08869841627912843901530571541, 7.15500795569719800791717256822, 7.69102384465031280252182372834, 8.061682512454208525152312623585, 8.181018576289516401880431443991, 8.526696157553920822526586781070, 9.484350459542729175077605349481
