L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s − 4·11-s + 6·12-s + 5·16-s − 8·17-s + 6·18-s − 8·19-s − 8·22-s − 12·23-s + 8·24-s + 4·27-s + 8·29-s − 12·31-s + 6·32-s − 8·33-s − 16·34-s + 9·36-s − 8·37-s − 16·38-s + 4·41-s − 4·43-s − 12·44-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s − 1.20·11-s + 1.73·12-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 1.83·19-s − 1.70·22-s − 2.50·23-s + 1.63·24-s + 0.769·27-s + 1.48·29-s − 2.15·31-s + 1.06·32-s − 1.39·33-s − 2.74·34-s + 3/2·36-s − 1.31·37-s − 2.59·38-s + 0.624·41-s − 0.609·43-s − 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 88 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.69448760872775431424330624088, −7.50126380372864451004065470575, −6.80833715568616776066162151718, −6.65629407270061499537416067959, −6.23842567236345614791146785580, −6.22935825226285093271029929896, −5.35910037404707384005628595880, −5.35064218028725671803783696695, −4.67956150575055488168313288148, −4.44345868849569532157017353077, −4.15595320720763145372097341428, −3.89213270331882715943269027458, −3.20665308908475193825144070662, −3.13999479075993403204083811994, −2.36480568303657116262803294499, −2.31793932927312864603746475987, −1.73041078594936187790446366695, −1.68743593710522068391541310608, 0, 0,
1.68743593710522068391541310608, 1.73041078594936187790446366695, 2.31793932927312864603746475987, 2.36480568303657116262803294499, 3.13999479075993403204083811994, 3.20665308908475193825144070662, 3.89213270331882715943269027458, 4.15595320720763145372097341428, 4.44345868849569532157017353077, 4.67956150575055488168313288148, 5.35064218028725671803783696695, 5.35910037404707384005628595880, 6.22935825226285093271029929896, 6.23842567236345614791146785580, 6.65629407270061499537416067959, 6.80833715568616776066162151718, 7.50126380372864451004065470575, 7.69448760872775431424330624088