| L(s) = 1 | − 8·7-s + 4·13-s − 16-s + 8·25-s − 16·31-s + 4·37-s − 32·43-s + 32·49-s + 32·61-s + 16·67-s + 4·73-s − 32·91-s − 44·97-s + 40·103-s + 8·112-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 1.10·13-s − 1/4·16-s + 8/5·25-s − 2.87·31-s + 0.657·37-s − 4.87·43-s + 32/7·49-s + 4.09·61-s + 1.95·67-s + 0.468·73-s − 3.35·91-s − 4.46·97-s + 3.94·103-s + 0.755·112-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3550379699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3550379699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 3518 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−12.16949045852122741856206588635, −11.39070334637959316061896944773, −11.32065327001559831372721487565, −11.07648150369747908571594140616, −10.73402037157714484917951818189, −9.993211947347321831766557155915, −9.985583919322433016490765967424, −9.813601344157596459911875113709, −9.523530201797702459658885758880, −8.989915349608132040349712194177, −8.674279076147765457799089192129, −8.388180912004417505596354636959, −8.200621657208690564393827562023, −7.13559581725033654267931037680, −7.09039865126819512771646727198, −6.76997249157825406892103851238, −6.56961946982146046531103976405, −6.05555100337708750065849810943, −5.61024030463355534518954703534, −5.26059532998057285416782806393, −4.62890793123549973374705059312, −3.66957350149866266741830980981, −3.42056620703233254502636074303, −3.37217461998631009762624239828, −2.25947599794642124177219078867,
2.25947599794642124177219078867, 3.37217461998631009762624239828, 3.42056620703233254502636074303, 3.66957350149866266741830980981, 4.62890793123549973374705059312, 5.26059532998057285416782806393, 5.61024030463355534518954703534, 6.05555100337708750065849810943, 6.56961946982146046531103976405, 6.76997249157825406892103851238, 7.09039865126819512771646727198, 7.13559581725033654267931037680, 8.200621657208690564393827562023, 8.388180912004417505596354636959, 8.674279076147765457799089192129, 8.989915349608132040349712194177, 9.523530201797702459658885758880, 9.813601344157596459911875113709, 9.985583919322433016490765967424, 9.993211947347321831766557155915, 10.73402037157714484917951818189, 11.07648150369747908571594140616, 11.32065327001559831372721487565, 11.39070334637959316061896944773, 12.16949045852122741856206588635
