| L(s) = 1 | − 24·17-s − 2·25-s + 32·41-s + 8·49-s − 24·73-s + 24·89-s + 40·97-s + 40·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.82·17-s − 2/5·25-s + 4.99·41-s + 8/7·49-s − 2.80·73-s + 2.54·89-s + 4.06·97-s + 3.76·113-s − 1.81·121-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 + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.164692366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.164692366\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−5.84694453977664795763714933248, −5.63715642687426188240211739647, −5.14465781847026199350209162475, −5.10992159159951469667188324035, −4.98318875561479592609496373966, −4.55877529660135868285585883638, −4.46176179718632249754436629867, −4.30729269598832855186505704992, −4.26710657276399351600561317578, −4.26158011759819995701475263881, −3.82647167758009548740217404851, −3.57891914098657557506911066651, −3.48445117736586501386896920363, −2.97941781867715517589090715590, −2.72493501628405493348547018346, −2.60456147691999933657454391239, −2.59409602731575447451225997185, −2.07663801992378300628937024696, −2.01661510915167464364584737480, −1.97600694214068064526719891345, −1.71113141720879485598061725570, −1.09372770515835110352837911073, −0.67146474165776443736312180241, −0.64256300515633003716405082391, −0.24460699787523916887487702126,
0.24460699787523916887487702126, 0.64256300515633003716405082391, 0.67146474165776443736312180241, 1.09372770515835110352837911073, 1.71113141720879485598061725570, 1.97600694214068064526719891345, 2.01661510915167464364584737480, 2.07663801992378300628937024696, 2.59409602731575447451225997185, 2.60456147691999933657454391239, 2.72493501628405493348547018346, 2.97941781867715517589090715590, 3.48445117736586501386896920363, 3.57891914098657557506911066651, 3.82647167758009548740217404851, 4.26158011759819995701475263881, 4.26710657276399351600561317578, 4.30729269598832855186505704992, 4.46176179718632249754436629867, 4.55877529660135868285585883638, 4.98318875561479592609496373966, 5.10992159159951469667188324035, 5.14465781847026199350209162475, 5.63715642687426188240211739647, 5.84694453977664795763714933248
