| L(s) = 1 | − 3-s + 5-s − 5·7-s + 3·9-s − 2·11-s − 15-s − 4·17-s + 2·19-s + 5·21-s − 23-s − 8·27-s + 18·29-s − 4·31-s + 2·33-s − 5·35-s − 4·37-s + 2·41-s + 18·43-s + 3·45-s + 18·49-s + 4·51-s + 10·53-s − 2·55-s − 2·57-s + 10·59-s − 9·61-s − 15·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.88·7-s + 9-s − 0.603·11-s − 0.258·15-s − 0.970·17-s + 0.458·19-s + 1.09·21-s − 0.208·23-s − 1.53·27-s + 3.34·29-s − 0.718·31-s + 0.348·33-s − 0.845·35-s − 0.657·37-s + 0.312·41-s + 2.74·43-s + 0.447·45-s + 18/7·49-s + 0.560·51-s + 1.37·53-s − 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.15·61-s − 1.88·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9534785697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9534785697\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−12.24422165925907341851244897416, −11.79460291339202189941915277972, −10.92202508336561741962648213581, −10.62121449087915718973300606748, −10.27163361767837742736937662996, −9.630427545704760159689836614916, −9.567019877068292669335617643125, −8.865511007557193034276711723516, −8.361505793328924693910925457053, −7.48073461747122807341826914782, −7.11272494485200689354641110841, −6.62122539626389563929799094781, −6.15057758984233623909254845878, −5.72469742834559554874646809618, −5.04160543654805963469012000655, −4.28899712693718974078982480025, −3.78086943524880099446214105900, −2.84302711075337274392253178577, −2.30683945843488954610806465251, −0.77596960107209571803117472311,
0.77596960107209571803117472311, 2.30683945843488954610806465251, 2.84302711075337274392253178577, 3.78086943524880099446214105900, 4.28899712693718974078982480025, 5.04160543654805963469012000655, 5.72469742834559554874646809618, 6.15057758984233623909254845878, 6.62122539626389563929799094781, 7.11272494485200689354641110841, 7.48073461747122807341826914782, 8.361505793328924693910925457053, 8.865511007557193034276711723516, 9.567019877068292669335617643125, 9.630427545704760159689836614916, 10.27163361767837742736937662996, 10.62121449087915718973300606748, 10.92202508336561741962648213581, 11.79460291339202189941915277972, 12.24422165925907341851244897416
