| L(s) = 1 | − 42·3-s − 4·7-s + 1.03e3·9-s − 5.90e3·13-s − 1.05e4·19-s + 168·21-s + 2.45e3·25-s − 1.28e4·27-s − 4.57e4·31-s + 6.81e4·37-s + 2.47e5·39-s + 1.28e4·43-s − 2.35e5·49-s + 4.41e5·57-s − 1.25e5·61-s − 4.14e3·63-s − 8.77e5·67-s − 1.46e6·73-s − 1.02e5·75-s − 6.81e5·79-s − 2.14e5·81-s + 2.36e4·91-s + 1.92e6·93-s − 5.62e5·97-s + 1.73e6·103-s + 1.30e6·109-s − 2.86e6·111-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 0.0116·7-s + 1.41·9-s − 2.68·13-s − 1.53·19-s + 0.0181·21-s + 0.156·25-s − 0.652·27-s − 1.53·31-s + 1.34·37-s + 4.17·39-s + 0.161·43-s − 1.99·49-s + 2.38·57-s − 0.551·61-s − 0.0165·63-s − 2.91·67-s − 3.75·73-s − 0.243·75-s − 1.38·79-s − 0.404·81-s + 0.0313·91-s + 2.39·93-s − 0.615·97-s + 1.58·103-s + 1.00·109-s − 2.09·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.04803302131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04803302131\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 14 p T + p^{6} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 98 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3541970 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2950 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 28202690 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5258 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 191004770 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1184779442 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 22898 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34058 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9219079010 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6406 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10801249342 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7253988050 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22449655150 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 62566 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 438698 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 251546372642 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 730510 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 340562 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 407613512306 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 844406214050 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 281086 T + p^{6} T^{2} )^{2} \) |
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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
−14.97781675895172901733639878350, −14.37691634881307594006368475625, −13.23562632078022806002230576974, −12.59177331733055821953997339945, −12.54812488052901975274724271200, −11.65222106110824888975741350442, −11.35407272411280269695587433553, −10.49095844248622571703129353979, −10.13626567086374785707868399141, −9.469413394961391305252114424009, −8.650526912619489569475340799901, −7.35746992061375402947811362799, −7.34124134284450123238853323853, −6.23783819092060161334909430956, −5.72612806215471172906465934690, −4.61319670456226779726741460731, −4.59474599525089693425096431441, −2.82929026465393684386746883517, −1.71985643325594034845848268691, −0.10545650039254847364174801472,
0.10545650039254847364174801472, 1.71985643325594034845848268691, 2.82929026465393684386746883517, 4.59474599525089693425096431441, 4.61319670456226779726741460731, 5.72612806215471172906465934690, 6.23783819092060161334909430956, 7.34124134284450123238853323853, 7.35746992061375402947811362799, 8.650526912619489569475340799901, 9.469413394961391305252114424009, 10.13626567086374785707868399141, 10.49095844248622571703129353979, 11.35407272411280269695587433553, 11.65222106110824888975741350442, 12.54812488052901975274724271200, 12.59177331733055821953997339945, 13.23562632078022806002230576974, 14.37691634881307594006368475625, 14.97781675895172901733639878350
