| L(s) = 1 | − 2·5-s + 9-s − 4·13-s + 12·17-s + 3·25-s + 4·29-s − 20·37-s + 4·41-s − 2·45-s − 14·49-s + 4·53-s + 4·61-s + 8·65-s − 28·73-s + 81-s − 24·85-s − 28·89-s + 4·97-s + 20·101-s + 20·109-s − 20·113-s − 4·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 1.10·13-s + 2.91·17-s + 3/5·25-s + 0.742·29-s − 3.28·37-s + 0.624·41-s − 0.298·45-s − 2·49-s + 0.549·53-s + 0.512·61-s + 0.992·65-s − 3.27·73-s + 1/9·81-s − 2.60·85-s − 2.96·89-s + 0.406·97-s + 1.99·101-s + 1.91·109-s − 1.88·113-s − 0.369·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.88407908599291033528419252210, −7.50925237905278923935250052131, −7.08681504004670289911753653803, −6.93140021140816704544563669625, −6.11496655716274741730349138464, −5.55946263088363195197157204381, −5.27165295209483690935777776364, −4.73356037048830854238430803855, −4.28889632980812263177388675758, −3.51487045961044115415176895273, −3.29264628324513736767476951185, −2.78204474347085703568577520578, −1.75820205155567607265884637873, −1.14288269174320867802245410197, 0,
1.14288269174320867802245410197, 1.75820205155567607265884637873, 2.78204474347085703568577520578, 3.29264628324513736767476951185, 3.51487045961044115415176895273, 4.28889632980812263177388675758, 4.73356037048830854238430803855, 5.27165295209483690935777776364, 5.55946263088363195197157204381, 6.11496655716274741730349138464, 6.93140021140816704544563669625, 7.08681504004670289911753653803, 7.50925237905278923935250052131, 7.88407908599291033528419252210
