| L(s) = 1 | − 16·19-s − 10·25-s + 4·31-s + 16·37-s − 32·103-s − 16·109-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3.67·19-s − 2·25-s + 0.718·31-s + 2.63·37-s − 3.15·103-s − 1.53·109-s + 1.27·121-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 + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431218129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431218129\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−5.63045393954638172188494618209, −5.49132887716754456403200517522, −5.16888927474375648083528463706, −4.96393884382612474219096136768, −4.83086731547623608003582137870, −4.60420425460600731763959549275, −4.29437340775562128654432464710, −4.18798816776879747443436082645, −4.11394668414404774472396847947, −4.00140658454423579610073785112, −3.77460678352694651819785755503, −3.66153651414517251206560734451, −3.16843187068028648422359091220, −2.87842304912992421571141283935, −2.73353966047190483946519546726, −2.68610099673332523804600045851, −2.51682842873814920428094647145, −2.05792504553396729337691159715, −1.78078014285196046920368611178, −1.76725165559088633346176531435, −1.75849633751597617554734064981, −1.07619896805680273018663696559, −0.829012333654465463349074760160, −0.47793205240919484970007725540, −0.19303620294808033317587379432,
0.19303620294808033317587379432, 0.47793205240919484970007725540, 0.829012333654465463349074760160, 1.07619896805680273018663696559, 1.75849633751597617554734064981, 1.76725165559088633346176531435, 1.78078014285196046920368611178, 2.05792504553396729337691159715, 2.51682842873814920428094647145, 2.68610099673332523804600045851, 2.73353966047190483946519546726, 2.87842304912992421571141283935, 3.16843187068028648422359091220, 3.66153651414517251206560734451, 3.77460678352694651819785755503, 4.00140658454423579610073785112, 4.11394668414404774472396847947, 4.18798816776879747443436082645, 4.29437340775562128654432464710, 4.60420425460600731763959549275, 4.83086731547623608003582137870, 4.96393884382612474219096136768, 5.16888927474375648083528463706, 5.49132887716754456403200517522, 5.63045393954638172188494618209
