| L(s) = 1 | − 2·4-s − 2·9-s + 3·16-s + 24·29-s + 16·31-s + 4·36-s − 8·41-s + 12·49-s + 32·59-s + 24·61-s − 4·64-s − 32·79-s + 3·81-s + 40·89-s + 8·101-s − 24·109-s − 48·116-s + 20·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 3/4·16-s + 4.45·29-s + 2.87·31-s + 2/3·36-s − 1.24·41-s + 12/7·49-s + 4.16·59-s + 3.07·61-s − 1/2·64-s − 3.60·79-s + 1/3·81-s + 4.23·89-s + 0.796·101-s − 2.29·109-s − 4.45·116-s + 1.81·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.088002344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.088002344\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 20 T^{2} - 4554 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 40166 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43640408334463485610490584492, −6.25937630540701701760101101182, −6.21796282018900067536929243605, −5.92726901526166188345707329438, −5.70414302346831728075242599817, −5.26676379532517021519491855326, −5.23232672530657350033399889749, −5.13420177029573658923327400288, −4.78491858075126431032517320221, −4.58076593078077721451022063210, −4.45144669092890016851131729770, −4.12634782922411667246438070605, −4.07330498168643432907503184240, −3.62452457784166558281009349183, −3.43462388299793915721786035978, −3.21557648644940086338065103100, −2.90517869699263197648725951207, −2.50691721364991197265604180035, −2.47170878743080326435062176622, −2.37799325256245014591943106011, −1.85060082777297354029388737562, −1.15105318031129360090332923900, −1.02650957668026914529034666969, −0.76825043146182310805696085172, −0.50716841142991648408298905213,
0.50716841142991648408298905213, 0.76825043146182310805696085172, 1.02650957668026914529034666969, 1.15105318031129360090332923900, 1.85060082777297354029388737562, 2.37799325256245014591943106011, 2.47170878743080326435062176622, 2.50691721364991197265604180035, 2.90517869699263197648725951207, 3.21557648644940086338065103100, 3.43462388299793915721786035978, 3.62452457784166558281009349183, 4.07330498168643432907503184240, 4.12634782922411667246438070605, 4.45144669092890016851131729770, 4.58076593078077721451022063210, 4.78491858075126431032517320221, 5.13420177029573658923327400288, 5.23232672530657350033399889749, 5.26676379532517021519491855326, 5.70414302346831728075242599817, 5.92726901526166188345707329438, 6.21796282018900067536929243605, 6.25937630540701701760101101182, 6.43640408334463485610490584492
