| L(s) = 1 | + 20·25-s − 22·49-s + 68·73-s − 20·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s − 3.14·49-s + 7.95·73-s − 2.03·97-s + 4·121-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 + 3.53·169-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.451650957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.451650957\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
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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
−6.79406965166221887969949098323, −6.46793066856069550189606681203, −6.28948264343811938105833470180, −6.06037773090367456905279098070, −5.79895799472135257364139125355, −5.51994544852649503507864242575, −5.13211595768645758965032608384, −5.11083587173936887938683770236, −5.04779733024366273939123354785, −4.59548474466280092342475655965, −4.58134959694985719040420210791, −4.41084653230360678798007462069, −3.91777257187625406307865820586, −3.59858725927547907429093926267, −3.37972452626993171615754740985, −3.28875549578341305780087299540, −3.14514390406080205955181449968, −2.71104094898758633722727685173, −2.30603590690326646081458245803, −2.23853284994544826539669118115, −1.99128604064616779946479360110, −1.37310772760674899507851260408, −1.07809220747594361840678521613, −0.921046181872475339632389313767, −0.37475913513593105720857592842,
0.37475913513593105720857592842, 0.921046181872475339632389313767, 1.07809220747594361840678521613, 1.37310772760674899507851260408, 1.99128604064616779946479360110, 2.23853284994544826539669118115, 2.30603590690326646081458245803, 2.71104094898758633722727685173, 3.14514390406080205955181449968, 3.28875549578341305780087299540, 3.37972452626993171615754740985, 3.59858725927547907429093926267, 3.91777257187625406307865820586, 4.41084653230360678798007462069, 4.58134959694985719040420210791, 4.59548474466280092342475655965, 5.04779733024366273939123354785, 5.11083587173936887938683770236, 5.13211595768645758965032608384, 5.51994544852649503507864242575, 5.79895799472135257364139125355, 6.06037773090367456905279098070, 6.28948264343811938105833470180, 6.46793066856069550189606681203, 6.79406965166221887969949098323
