| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s − 8·11-s + 4·15-s − 10·17-s − 10·19-s + 4·21-s + 2·23-s − 2·25-s + 4·27-s − 4·31-s − 16·33-s + 4·35-s − 4·41-s − 2·43-s + 6·45-s − 8·47-s − 6·49-s − 20·51-s + 6·53-s − 16·55-s − 20·57-s − 8·59-s + 6·63-s − 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 2.41·11-s + 1.03·15-s − 2.42·17-s − 2.29·19-s + 0.872·21-s + 0.417·23-s − 2/5·25-s + 0.769·27-s − 0.718·31-s − 2.78·33-s + 0.676·35-s − 0.624·41-s − 0.304·43-s + 0.894·45-s − 1.16·47-s − 6/7·49-s − 2.80·51-s + 0.824·53-s − 2.15·55-s − 2.64·57-s − 1.04·59-s + 0.755·63-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.115997781253412285452799498290, −8.084051926373038396915497381252, −7.50248475251985106727903395171, −7.20168892735224606665913517515, −6.67169948047704505004965306015, −6.51165796321878820127652972834, −5.90514808461132253692573244477, −5.68658064749301217449407748682, −4.92608224608797885293664641535, −4.89226018923646975551211011261, −4.45972247054425445286630527945, −4.12259063501064428436824062782, −3.50872506551610863095515269526, −2.88091064853498118821036685962, −2.55691344686327950583785915245, −2.27377589777439877721660510282, −1.75020256190796892069954039788, −1.66213090439218947759706895598, 0, 0,
1.66213090439218947759706895598, 1.75020256190796892069954039788, 2.27377589777439877721660510282, 2.55691344686327950583785915245, 2.88091064853498118821036685962, 3.50872506551610863095515269526, 4.12259063501064428436824062782, 4.45972247054425445286630527945, 4.89226018923646975551211011261, 4.92608224608797885293664641535, 5.68658064749301217449407748682, 5.90514808461132253692573244477, 6.51165796321878820127652972834, 6.67169948047704505004965306015, 7.20168892735224606665913517515, 7.50248475251985106727903395171, 8.084051926373038396915497381252, 8.115997781253412285452799498290
