| L(s) = 1 | − 4·3-s − 2·5-s + 6·9-s − 3·11-s + 4·13-s + 8·15-s + 7·17-s − 2·19-s − 7·25-s + 4·27-s − 8·29-s − 8·31-s + 12·33-s + 4·37-s − 16·39-s + 6·41-s − 2·43-s − 12·45-s + 3·47-s − 28·51-s + 2·53-s + 6·55-s + 8·57-s − 8·59-s + 17·61-s − 8·65-s + 6·71-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.894·5-s + 2·9-s − 0.904·11-s + 1.10·13-s + 2.06·15-s + 1.69·17-s − 0.458·19-s − 7/5·25-s + 0.769·27-s − 1.48·29-s − 1.43·31-s + 2.08·33-s + 0.657·37-s − 2.56·39-s + 0.937·41-s − 0.304·43-s − 1.78·45-s + 0.437·47-s − 3.92·51-s + 0.274·53-s + 0.809·55-s + 1.05·57-s − 1.04·59-s + 2.17·61-s − 0.992·65-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15 T + 188 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 125 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 138 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.66772236376216417950722761979, −7.41844493327151496567885410687, −6.92928575005735506097120060323, −6.61684898577665528235642997781, −6.11595226030484875521503419561, −5.79792728434284334003909875474, −5.66423659455369253905440290275, −5.54802738956672875082838003953, −4.87326485324748294252855286998, −4.84320561306404727131219965527, −4.10863472142883553969771089425, −3.78345626724634685819771973419, −3.53817318310079490295122797449, −3.13012576482682043869654226361, −2.28638946275807995511729319729, −2.04559235537100395995222611040, −1.06523769797187141954430573575, −0.925657262824924561006678775950, 0, 0,
0.925657262824924561006678775950, 1.06523769797187141954430573575, 2.04559235537100395995222611040, 2.28638946275807995511729319729, 3.13012576482682043869654226361, 3.53817318310079490295122797449, 3.78345626724634685819771973419, 4.10863472142883553969771089425, 4.84320561306404727131219965527, 4.87326485324748294252855286998, 5.54802738956672875082838003953, 5.66423659455369253905440290275, 5.79792728434284334003909875474, 6.11595226030484875521503419561, 6.61684898577665528235642997781, 6.92928575005735506097120060323, 7.41844493327151496567885410687, 7.66772236376216417950722761979
