| L(s) = 1 | + 8·5-s − 4·13-s + 4·17-s + 38·25-s − 16·29-s − 20·37-s − 8·41-s − 28·53-s − 32·65-s + 28·73-s + 18·81-s + 32·85-s − 28·97-s − 48·109-s + 36·113-s − 20·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 1.10·13-s + 0.970·17-s + 38/5·25-s − 2.97·29-s − 3.28·37-s − 1.24·41-s − 3.84·53-s − 3.96·65-s + 3.27·73-s + 2·81-s + 3.47·85-s − 2.84·97-s − 4.59·109-s + 3.38·113-s − 1.81·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460927437\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 542 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3326 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.70447746767751334270346422246, −6.56092404051223673078947372650, −6.55066560635412489296890055329, −6.36958317014230045583027012553, −5.90778790918733634070926631841, −5.77233802754215129801695055420, −5.48973860372372011744900298779, −5.36384834673605009816578433895, −5.18792158427783697959793701449, −5.16915417877838436444398136043, −4.82683924462730180611968558361, −4.67693418926345662381358637912, −4.21059687140627401186053286543, −3.67030678293651480734874533900, −3.65396983861386526050381774564, −3.33255254272453614835478172189, −3.15169424762984126071456811197, −2.69345042220394502671205957342, −2.42854529908805877705980196381, −2.24096494037619119545903925712, −1.82968493364292697137694916842, −1.75791555210869630123897030211, −1.39638625675540828333485341571, −1.28010530310604080380166095018, −0.25121060942101382308634355860,
0.25121060942101382308634355860, 1.28010530310604080380166095018, 1.39638625675540828333485341571, 1.75791555210869630123897030211, 1.82968493364292697137694916842, 2.24096494037619119545903925712, 2.42854529908805877705980196381, 2.69345042220394502671205957342, 3.15169424762984126071456811197, 3.33255254272453614835478172189, 3.65396983861386526050381774564, 3.67030678293651480734874533900, 4.21059687140627401186053286543, 4.67693418926345662381358637912, 4.82683924462730180611968558361, 5.16915417877838436444398136043, 5.18792158427783697959793701449, 5.36384834673605009816578433895, 5.48973860372372011744900298779, 5.77233802754215129801695055420, 5.90778790918733634070926631841, 6.36958317014230045583027012553, 6.55066560635412489296890055329, 6.56092404051223673078947372650, 6.70447746767751334270346422246
