| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s − 7·13-s − 7·17-s − 6·19-s + 6·23-s − 7·25-s − 29-s − 8·31-s + 4·35-s − 37-s + 9·41-s + 6·43-s + 12·47-s + 7·49-s + 18·53-s + 4·55-s − 61-s − 14·65-s − 2·67-s − 6·71-s + 22·73-s + 4·77-s + 8·79-s − 28·83-s − 14·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 1.94·13-s − 1.69·17-s − 1.37·19-s + 1.25·23-s − 7/5·25-s − 0.185·29-s − 1.43·31-s + 0.676·35-s − 0.164·37-s + 1.40·41-s + 0.914·43-s + 1.75·47-s + 49-s + 2.47·53-s + 0.539·55-s − 0.128·61-s − 1.73·65-s − 0.244·67-s − 0.712·71-s + 2.57·73-s + 0.455·77-s + 0.900·79-s − 3.07·83-s − 1.51·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907017172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907017172\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.492359838557311160501539657213, −9.077954766469319073780637007453, −8.684164390808906012298456533420, −8.464851379723652072079705300118, −7.73708108933644115153815071972, −7.30958705643368048550068711356, −7.17307595524071626919888044507, −6.76463081804535703313371604536, −6.20681328395694177563685806825, −5.70594146693258427351759253874, −5.47746960808328299069629338433, −5.04456396530701919575016481019, −4.32031345527362682821274598328, −4.21853369469580555861462839887, −3.85147281724436259496278128083, −2.70981605593896303722605111542, −2.26997847929661663948090435957, −2.25500474957632478986561890813, −1.51218474612191932939092501224, −0.48468420348916951737914194887,
0.48468420348916951737914194887, 1.51218474612191932939092501224, 2.25500474957632478986561890813, 2.26997847929661663948090435957, 2.70981605593896303722605111542, 3.85147281724436259496278128083, 4.21853369469580555861462839887, 4.32031345527362682821274598328, 5.04456396530701919575016481019, 5.47746960808328299069629338433, 5.70594146693258427351759253874, 6.20681328395694177563685806825, 6.76463081804535703313371604536, 7.17307595524071626919888044507, 7.30958705643368048550068711356, 7.73708108933644115153815071972, 8.464851379723652072079705300118, 8.684164390808906012298456533420, 9.077954766469319073780637007453, 9.492359838557311160501539657213
