| L(s) = 1 | + 4·5-s + 6·13-s + 8·17-s + 5·25-s − 4·29-s + 2·37-s + 16·41-s − 14·49-s + 8·53-s + 10·61-s + 24·65-s − 6·73-s + 32·85-s + 16·89-s − 36·97-s + 40·101-s + 6·109-s + 16·113-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.66·13-s + 1.94·17-s + 25-s − 0.742·29-s + 0.328·37-s + 2.49·41-s − 2·49-s + 1.09·53-s + 1.28·61-s + 2.97·65-s − 0.702·73-s + 3.47·85-s + 1.69·89-s − 3.65·97-s + 3.98·101-s + 0.574·109-s + 1.50·113-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.292520510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.292520510\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−8.994210977633164635761506859403, −8.924617339451835593653033504269, −8.274485677124726335553975472353, −7.987305659786083206758074514462, −7.53993379307156996093139778832, −7.27009498010490530718143265202, −6.44488767609829650456319452562, −6.39474368644208140190466486211, −5.89902613717607467979422048431, −5.65676368469135902964853806025, −5.40954846759182740913614781329, −4.92110471001131665974461053655, −4.13401481753795826376455251387, −3.95727081800436517263191744320, −3.14307793317028215606377158882, −3.11397351229638985652968555255, −2.16936907008936543068345935141, −1.93244442607624280792188888914, −1.20945780797206098361271470292, −0.870512331980832436414106425658,
0.870512331980832436414106425658, 1.20945780797206098361271470292, 1.93244442607624280792188888914, 2.16936907008936543068345935141, 3.11397351229638985652968555255, 3.14307793317028215606377158882, 3.95727081800436517263191744320, 4.13401481753795826376455251387, 4.92110471001131665974461053655, 5.40954846759182740913614781329, 5.65676368469135902964853806025, 5.89902613717607467979422048431, 6.39474368644208140190466486211, 6.44488767609829650456319452562, 7.27009498010490530718143265202, 7.53993379307156996093139778832, 7.987305659786083206758074514462, 8.274485677124726335553975472353, 8.924617339451835593653033504269, 8.994210977633164635761506859403
