| L(s) = 1 | + 9-s + 2·25-s + 6·41-s − 4·49-s + 2·73-s − 2·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
| L(s) = 1 | + 9-s + 2·25-s + 6·41-s − 4·49-s + 2·73-s − 2·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.326388918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326388918\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−6.29410448524323304770776831846, −5.89644652236764756106245799314, −5.86629185944619482872825990953, −5.71770143118230883424237242428, −5.54001269221690852372945713596, −5.08055256612131778026644402121, −5.07153409069329517340915194154, −4.64534483487885840530721525383, −4.63112516001289953972194919901, −4.51069996493852308299678953803, −4.21986417870853949105608461912, −4.13487674822379063142720795821, −3.84253095331439363000355724291, −3.35523064947248809178457178706, −3.35435615715745318402713350051, −3.22152514804833536069259133054, −2.86952313007425926384788782466, −2.55801760219232205191435546537, −2.43156803345714958681548690246, −2.05536133869214462569544671309, −2.02190435125771342779680091526, −1.39285572197834043967516395348, −1.25405510463981728624149528695, −0.985970076253512030964467383754, −0.64810164510278354328469199958,
0.64810164510278354328469199958, 0.985970076253512030964467383754, 1.25405510463981728624149528695, 1.39285572197834043967516395348, 2.02190435125771342779680091526, 2.05536133869214462569544671309, 2.43156803345714958681548690246, 2.55801760219232205191435546537, 2.86952313007425926384788782466, 3.22152514804833536069259133054, 3.35435615715745318402713350051, 3.35523064947248809178457178706, 3.84253095331439363000355724291, 4.13487674822379063142720795821, 4.21986417870853949105608461912, 4.51069996493852308299678953803, 4.63112516001289953972194919901, 4.64534483487885840530721525383, 5.07153409069329517340915194154, 5.08055256612131778026644402121, 5.54001269221690852372945713596, 5.71770143118230883424237242428, 5.86629185944619482872825990953, 5.89644652236764756106245799314, 6.29410448524323304770776831846
