| L(s) = 1 | − 3-s + 9-s + 4·17-s − 8·19-s − 25-s − 27-s + 8·41-s − 8·43-s − 2·49-s − 4·51-s + 8·57-s − 16·59-s + 8·67-s − 4·73-s + 75-s + 81-s − 8·83-s + 8·89-s − 28·97-s + 8·107-s − 20·113-s − 18·121-s − 8·123-s + 127-s + 8·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.970·17-s − 1.83·19-s − 1/5·25-s − 0.192·27-s + 1.24·41-s − 1.21·43-s − 2/7·49-s − 0.560·51-s + 1.05·57-s − 2.08·59-s + 0.977·67-s − 0.468·73-s + 0.115·75-s + 1/9·81-s − 0.878·83-s + 0.847·89-s − 2.84·97-s + 0.773·107-s − 1.88·113-s − 1.63·121-s − 0.721·123-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.346291720121995717712093975596, −8.124319318961205390000648166586, −7.61948249871416466715769020252, −7.01807698922579834014571363934, −6.59751880432142696160633479249, −6.14527405832827378577925110928, −5.69478256156717491270165659232, −5.21106317421538041180495208875, −4.51900425270164728349791276127, −4.21343282699377366119345913002, −3.50903017322962956121673785600, −2.83664421549346698687301375081, −2.05129353552081938000857796892, −1.27841005985528659821354329133, 0,
1.27841005985528659821354329133, 2.05129353552081938000857796892, 2.83664421549346698687301375081, 3.50903017322962956121673785600, 4.21343282699377366119345913002, 4.51900425270164728349791276127, 5.21106317421538041180495208875, 5.69478256156717491270165659232, 6.14527405832827378577925110928, 6.59751880432142696160633479249, 7.01807698922579834014571363934, 7.61948249871416466715769020252, 8.124319318961205390000648166586, 8.346291720121995717712093975596
