| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s − 4·11-s + 4·13-s − 15-s − 16-s + 2·17-s − 4·19-s + 4·22-s + 8·23-s − 24-s − 4·26-s + 27-s + 12·29-s + 30-s − 8·31-s + 4·33-s − 2·34-s + 2·37-s + 4·38-s − 4·39-s + 40-s − 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.10·13-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s + 2.22·29-s + 0.182·30-s − 1.43·31-s + 0.696·33-s − 0.342·34-s + 0.328·37-s + 0.648·38-s − 0.640·39-s + 0.158·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6823371213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6823371213\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.978837855507344733657111264519, −9.113847329688132631774509124670, −8.987059752399097977986200468142, −8.374604616551651437812915639449, −8.309345438308214623539627249730, −7.899560923338268204413551607962, −7.26495924099365226487594576304, −6.67408382143044122223628453181, −6.65777622336347324527784514159, −6.12575486857870100073204073970, −5.39009275219658888685650062889, −5.32987490783674802645499227150, −4.76346787471198151281217965712, −4.41626881756221424194199488282, −3.55998383405255890869216238897, −3.07478786587717158918490466128, −2.73734592217225726954938329040, −1.65841641839971348941668591857, −1.46117459097085593644753593475, −0.39980701308505548198358754887,
0.39980701308505548198358754887, 1.46117459097085593644753593475, 1.65841641839971348941668591857, 2.73734592217225726954938329040, 3.07478786587717158918490466128, 3.55998383405255890869216238897, 4.41626881756221424194199488282, 4.76346787471198151281217965712, 5.32987490783674802645499227150, 5.39009275219658888685650062889, 6.12575486857870100073204073970, 6.65777622336347324527784514159, 6.67408382143044122223628453181, 7.26495924099365226487594576304, 7.899560923338268204413551607962, 8.309345438308214623539627249730, 8.374604616551651437812915639449, 8.987059752399097977986200468142, 9.113847329688132631774509124670, 9.978837855507344733657111264519
