| L(s) = 1 | + 2-s + 3-s + 6-s + 4·7-s − 8-s + 3·9-s + 6·13-s + 4·14-s − 16-s + 7·17-s + 3·18-s + 7·19-s + 4·21-s + 2·23-s − 24-s + 6·26-s + 8·27-s − 10·29-s − 4·31-s + 7·34-s − 8·37-s + 7·38-s + 6·39-s − 2·41-s + 4·42-s − 12·43-s + 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 9-s + 1.66·13-s + 1.06·14-s − 1/4·16-s + 1.69·17-s + 0.707·18-s + 1.60·19-s + 0.872·21-s + 0.417·23-s − 0.204·24-s + 1.17·26-s + 1.53·27-s − 1.85·29-s − 0.718·31-s + 1.20·34-s − 1.31·37-s + 1.13·38-s + 0.960·39-s − 0.312·41-s + 0.617·42-s − 1.82·43-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.723171739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.723171739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34599039577363938155291269900, −9.753896958038120181784811519343, −9.257148210482455452088647790531, −9.180718358012264093291878714398, −8.395764872650266971956110983382, −8.119250635477307103428660859525, −7.82417661959673157211801893168, −7.43396729172931931625275618412, −6.79156902407378647287619746137, −6.50209472958931857661344709605, −5.68040130349886059775860832322, −5.23211496517706792481320258497, −5.15097657793165521358766752593, −4.58659887787054676319870682825, −3.73999173967527726462976642243, −3.40874993968444127245205612980, −3.39247267870296325394236208298, −2.17986438884057206823037595684, −1.39439241757736951094852567552, −1.27015546854278142233877311079,
1.27015546854278142233877311079, 1.39439241757736951094852567552, 2.17986438884057206823037595684, 3.39247267870296325394236208298, 3.40874993968444127245205612980, 3.73999173967527726462976642243, 4.58659887787054676319870682825, 5.15097657793165521358766752593, 5.23211496517706792481320258497, 5.68040130349886059775860832322, 6.50209472958931857661344709605, 6.79156902407378647287619746137, 7.43396729172931931625275618412, 7.82417661959673157211801893168, 8.119250635477307103428660859525, 8.395764872650266971956110983382, 9.180718358012264093291878714398, 9.257148210482455452088647790531, 9.753896958038120181784811519343, 10.34599039577363938155291269900
