| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 2·7-s − 8-s − 10-s − 6·11-s + 13-s − 2·14-s + 15-s − 16-s − 3·17-s − 2·19-s + 2·21-s − 6·22-s + 24-s + 26-s + 27-s − 6·29-s + 30-s − 8·31-s + 6·33-s − 3·34-s + 2·35-s + 11·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 0.277·13-s − 0.534·14-s + 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.436·21-s − 1.27·22-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s + 1.04·33-s − 0.514·34-s + 0.338·35-s + 1.80·37-s − 0.324·38-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
| \(L(1)\) |
\(\approx\) |
\(0.06820570268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06820570268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 11 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.43128903120370529331587079626, −9.528598399655838776482392907883, −9.397475956183968375452256020224, −8.780544473229872773176726811034, −8.413891767465397692938730610832, −7.83911775193637718008306784699, −7.59516305255420728791166363044, −7.06407636279771638030903345979, −6.66182893324257744814483415743, −6.02259697832898749144219131149, −5.77346642013925864030653835475, −5.47147827991944199742244757752, −4.79779302550387108388619607525, −4.49089706044109309590591201849, −3.94455919470724546601306988437, −3.44398361314567617832430678531, −2.78970358859114631512576989945, −2.49018037182304158806041699831, −1.53349777772669168051512519892, −0.10303702758514352755249055477,
0.10303702758514352755249055477, 1.53349777772669168051512519892, 2.49018037182304158806041699831, 2.78970358859114631512576989945, 3.44398361314567617832430678531, 3.94455919470724546601306988437, 4.49089706044109309590591201849, 4.79779302550387108388619607525, 5.47147827991944199742244757752, 5.77346642013925864030653835475, 6.02259697832898749144219131149, 6.66182893324257744814483415743, 7.06407636279771638030903345979, 7.59516305255420728791166363044, 7.83911775193637718008306784699, 8.413891767465397692938730610832, 8.780544473229872773176726811034, 9.397475956183968375452256020224, 9.528598399655838776482392907883, 10.43128903120370529331587079626
