| L(s) = 1 | − 16·2-s + 192·4-s − 2.04e3·8-s + 208·9-s − 7.08e3·13-s + 2.04e4·16-s − 9.82e3·17-s − 3.32e3·18-s + 3.12e4·25-s + 1.13e5·26-s − 1.96e5·32-s + 1.57e5·34-s + 3.99e4·36-s + 1.98e5·49-s − 5.00e5·50-s − 1.36e6·52-s − 5.30e5·53-s + 1.83e6·64-s − 1.88e6·68-s − 4.25e5·72-s − 4.88e5·81-s − 1.88e6·89-s − 3.18e6·98-s + 6.00e6·100-s − 4.11e6·101-s + 1.45e7·104-s + 8.49e6·106-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 0.285·9-s − 3.22·13-s + 5·16-s − 2·17-s − 0.570·18-s + 2·25-s + 6.45·26-s − 6·32-s + 4·34-s + 0.855·36-s + 1.69·49-s − 4·50-s − 9.67·52-s − 3.56·53-s + 7·64-s − 6·68-s − 1.14·72-s − 0.918·81-s − 2.67·89-s − 3.38·98-s + 6·100-s − 3.99·101-s + 12.9·104-s + 7.12·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.08015303005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08015303005\) |
| \(L(4)\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 208 T^{2} + p^{12} T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 198848 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 137072 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3544 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 285674272 T^{2} + p^{12} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 692486912 T^{2} + p^{12} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 265354 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 108773480608 T^{2} + p^{12} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 291617928992 T^{2} + p^{12} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 944512 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−14.11868264586336809250001721274, −12.86167242186776186655704777782, −12.42235707272805975217001012362, −12.17440055912095601162219714775, −11.18341551013701982139704900388, −10.98870767601441902758950226272, −10.23380006479101459578943788004, −9.724360421279862978213479722312, −9.307022435477381113469942203551, −8.754824972496913039021678935915, −8.036204238643376721572065434445, −7.30877828663241940288664703279, −6.97627257033463276896208028675, −6.49150719721607815117053200131, −5.27318197710480259135024598619, −4.51373982904852819466100307971, −2.72750160075617850249069073737, −2.53421757018927879095549634602, −1.47467069917989408474053844707, −0.15046305173620593326099175162,
0.15046305173620593326099175162, 1.47467069917989408474053844707, 2.53421757018927879095549634602, 2.72750160075617850249069073737, 4.51373982904852819466100307971, 5.27318197710480259135024598619, 6.49150719721607815117053200131, 6.97627257033463276896208028675, 7.30877828663241940288664703279, 8.036204238643376721572065434445, 8.754824972496913039021678935915, 9.307022435477381113469942203551, 9.724360421279862978213479722312, 10.23380006479101459578943788004, 10.98870767601441902758950226272, 11.18341551013701982139704900388, 12.17440055912095601162219714775, 12.42235707272805975217001012362, 12.86167242186776186655704777782, 14.11868264586336809250001721274
