| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 10-s − 6·11-s + 5·13-s + 15-s − 16-s − 2·17-s + 5·19-s − 6·22-s + 10·23-s − 24-s + 5·26-s − 27-s + 12·29-s + 30-s − 12·31-s − 6·33-s − 2·34-s + 37-s + 5·38-s + 5·39-s − 40-s − 2·41-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 − 1.80·11-s + 1.38·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 1.14·19-s − 1.27·22-s + 2.08·23-s − 0.204·24-s + 0.980·26-s − 0.192·27-s + 2.22·29-s + 0.182·30-s − 2.15·31-s − 1.04·33-s − 0.342·34-s + 0.164·37-s + 0.811·38-s + 0.800·39-s − 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.822725763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.822725763\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 113 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 5 T - 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−10.18292189491926584971762581825, −9.451527764073938227669520685484, −9.206007835969312755106643406564, −8.778575515497031681308082004788, −8.575555120512153594599415289318, −7.952092881369026163692617986427, −7.48300381395437310381146618550, −7.33284778826714492784211051714, −6.43070490361064058370796256610, −6.38661684263454426976783781788, −5.52512989908406569372306227121, −5.43264075326392101073570722249, −4.84736269019168766151916501055, −4.58560061388893452527437944368, −3.65289029493915044012674541124, −3.40622043192681713601759837762, −2.76778029490586732208383663897, −2.54712041799946166243148096999, −1.60514424115448976317171268497, −0.77068668204527063927287674008,
0.77068668204527063927287674008, 1.60514424115448976317171268497, 2.54712041799946166243148096999, 2.76778029490586732208383663897, 3.40622043192681713601759837762, 3.65289029493915044012674541124, 4.58560061388893452527437944368, 4.84736269019168766151916501055, 5.43264075326392101073570722249, 5.52512989908406569372306227121, 6.38661684263454426976783781788, 6.43070490361064058370796256610, 7.33284778826714492784211051714, 7.48300381395437310381146618550, 7.952092881369026163692617986427, 8.575555120512153594599415289318, 8.778575515497031681308082004788, 9.206007835969312755106643406564, 9.451527764073938227669520685484, 10.18292189491926584971762581825
