| L(s) = 1 | − 6·2-s + 66·3-s + 18·4-s − 396·6-s + 414·7-s − 384·8-s + 2.17e3·9-s − 2.37e3·11-s + 1.18e3·12-s + 3.09e3·13-s − 2.48e3·14-s + 188·16-s + 6.50e3·17-s − 1.30e4·18-s + 2.73e4·21-s + 1.42e4·22-s + 1.06e4·23-s − 2.53e4·24-s − 1.85e4·26-s + 4.81e4·27-s + 7.45e3·28-s + 5.08e4·31-s + 2.34e4·32-s − 1.56e5·33-s − 3.90e4·34-s + 3.92e4·36-s + 4.11e4·37-s + ⋯ |
| L(s) = 1 | − 3/4·2-s + 22/9·3-s + 9/32·4-s − 1.83·6-s + 1.20·7-s − 3/4·8-s + 2.98·9-s − 1.78·11-s + 0.687·12-s + 1.40·13-s − 0.905·14-s + 0.0458·16-s + 1.32·17-s − 2.24·18-s + 2.95·21-s + 1.33·22-s + 0.873·23-s − 1.83·24-s − 1.05·26-s + 22/9·27-s + 0.339·28-s + 1.70·31-s + 0.715·32-s − 4.36·33-s − 0.992·34-s + 0.840·36-s + 0.813·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.446926238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446926238\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 p T + 9 p T^{2} + 3 p^{7} T^{3} + p^{12} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 22 p T + 242 p^{2} T^{2} - 22 p^{7} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 414 T + 85698 T^{2} - 414 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 108 p T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3096 T + 4792608 T^{2} - 3096 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6504 T + 21151008 T^{2} - 6504 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 68488162 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 462 p T + 106722 p^{2} T^{2} - 462 p^{7} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1110258542 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 25432 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 41184 T + 848060928 T^{2} - 41184 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 19008 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 160686 T + 12909995298 T^{2} - 160686 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 32274 T + 520805538 T^{2} - 32274 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 311784 T + 48604631328 T^{2} + 311784 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 45368565118 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 178112 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 480546 T + 115462229058 T^{2} + 480546 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 617328 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 613224 T + 188021837088 T^{2} + 613224 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 431997693442 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 268194 T + 35964010818 T^{2} + 268194 p^{6} T^{3} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 920916709022 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1621224 T + 1314183629088 T^{2} - 1621224 p^{6} T^{3} + p^{12} T^{4} \) |
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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
−16.32555895458355759607541726397, −15.69206753681040712152749835791, −15.35569884779034840784646924681, −14.52593140936259082187547541161, −14.36402878780748786139612691764, −13.36986212345423649390399418604, −13.29432065329406095777348440430, −12.11515071123496556142617795658, −11.16087702604561657483412551232, −10.41883953950850932757067104181, −9.677471511727883883232930875970, −8.759931847608313073744194070038, −8.558660131982835716953493745208, −7.896789407476087169817192159153, −7.51755370724934464855090395902, −5.88617417876145791382468430769, −4.51771608341138574302790855279, −3.06251685451911149565720320127, −2.65617465073747111338180535605, −1.26224054871118365188872180108,
1.26224054871118365188872180108, 2.65617465073747111338180535605, 3.06251685451911149565720320127, 4.51771608341138574302790855279, 5.88617417876145791382468430769, 7.51755370724934464855090395902, 7.896789407476087169817192159153, 8.558660131982835716953493745208, 8.759931847608313073744194070038, 9.677471511727883883232930875970, 10.41883953950850932757067104181, 11.16087702604561657483412551232, 12.11515071123496556142617795658, 13.29432065329406095777348440430, 13.36986212345423649390399418604, 14.36402878780748786139612691764, 14.52593140936259082187547541161, 15.35569884779034840784646924681, 15.69206753681040712152749835791, 16.32555895458355759607541726397
