| L(s) = 1 | − 4·4-s + 4·13-s + 10·16-s − 16·52-s − 20·64-s − 4·67-s − 81-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 40·208-s + 211-s + ⋯ |
| L(s) = 1 | − 4·4-s + 4·13-s + 10·16-s − 16·52-s − 20·64-s − 4·67-s − 81-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 40·208-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3451890348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3451890348\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−7.82956995653123268778773059071, −7.44260067109104452478358349948, −6.93877467487005068763566155171, −6.90733675735922229648144695972, −6.55463420897253909556409351222, −5.93226144079959852517820556361, −5.92577237915978534975049110222, −5.91396263915775707123898883969, −5.81377201354538384911422407052, −5.46220063401616106482853365663, −4.96110723840541831610336365924, −4.92373286025177504971815052936, −4.74143429003153392351442594367, −4.24280479494134072127724901382, −4.12965944799308626802165964252, −3.98667549045219346519891785970, −3.87004982682175354458229525103, −3.50458912913076591424491464664, −3.20605174806432004589495505949, −3.06410198624547711167492618776, −2.73495878606900186375558057308, −1.64694741879472401837446572265, −1.36230105764151789874643651832, −1.35190921737122512040792026496, −0.69164488747293383763982657862,
0.69164488747293383763982657862, 1.35190921737122512040792026496, 1.36230105764151789874643651832, 1.64694741879472401837446572265, 2.73495878606900186375558057308, 3.06410198624547711167492618776, 3.20605174806432004589495505949, 3.50458912913076591424491464664, 3.87004982682175354458229525103, 3.98667549045219346519891785970, 4.12965944799308626802165964252, 4.24280479494134072127724901382, 4.74143429003153392351442594367, 4.92373286025177504971815052936, 4.96110723840541831610336365924, 5.46220063401616106482853365663, 5.81377201354538384911422407052, 5.91396263915775707123898883969, 5.92577237915978534975049110222, 5.93226144079959852517820556361, 6.55463420897253909556409351222, 6.90733675735922229648144695972, 6.93877467487005068763566155171, 7.44260067109104452478358349948, 7.82956995653123268778773059071
