L(s) = 1 | + 2-s + 4-s + 8-s + 16-s + 6·19-s + 4·25-s + 8·29-s + 32-s + 6·38-s + 4·41-s − 10·49-s + 4·50-s − 4·53-s + 8·58-s − 12·59-s + 12·61-s + 64-s + 20·71-s + 4·73-s + 6·76-s + 4·82-s − 8·89-s − 10·98-s + 4·100-s − 4·106-s − 20·107-s − 36·113-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s + 1.37·19-s + 4/5·25-s + 1.48·29-s + 0.176·32-s + 0.973·38-s + 0.624·41-s − 1.42·49-s + 0.565·50-s − 0.549·53-s + 1.05·58-s − 1.56·59-s + 1.53·61-s + 1/8·64-s + 2.37·71-s + 0.468·73-s + 0.688·76-s + 0.441·82-s − 0.847·89-s − 1.01·98-s + 2/5·100-s − 0.388·106-s − 1.93·107-s − 3.38·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.281513779\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.281513779\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.321505648973296636788635200813, −8.080188314158818571210357862812, −7.70571790366317507184134389056, −7.01622164753124191835971065950, −6.58480449283042001488534802270, −6.43473537524499031486534765628, −5.55176025751812533688654781979, −5.26879507436222136477228335690, −4.82559343853086042764183285583, −4.23017731804730474872268638152, −3.68809427388098652750852431406, −2.95189892597634995482564705741, −2.74653438892517003147187744868, −1.71656917410509587158465868360, −0.922340966506990550757887924806,
0.922340966506990550757887924806, 1.71656917410509587158465868360, 2.74653438892517003147187744868, 2.95189892597634995482564705741, 3.68809427388098652750852431406, 4.23017731804730474872268638152, 4.82559343853086042764183285583, 5.26879507436222136477228335690, 5.55176025751812533688654781979, 6.43473537524499031486534765628, 6.58480449283042001488534802270, 7.01622164753124191835971065950, 7.70571790366317507184134389056, 8.080188314158818571210357862812, 8.321505648973296636788635200813