| L(s) = 1 | + 2·3-s − 4·4-s − 3·9-s − 8·12-s + 5·13-s + 12·16-s + 6·17-s − 12·23-s + 25-s − 14·27-s + 6·29-s + 12·36-s + 10·39-s − 20·43-s + 24·48-s + 49-s + 12·51-s − 20·52-s + 24·53-s + 16·61-s − 32·64-s − 24·68-s − 24·69-s + 2·75-s − 2·79-s − 4·81-s + 12·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s − 9-s − 2.30·12-s + 1.38·13-s + 3·16-s + 1.45·17-s − 2.50·23-s + 1/5·25-s − 2.69·27-s + 1.11·29-s + 2·36-s + 1.60·39-s − 3.04·43-s + 3.46·48-s + 1/7·49-s + 1.68·51-s − 2.77·52-s + 3.29·53-s + 2.04·61-s − 4·64-s − 2.91·68-s − 2.88·69-s + 0.230·75-s − 0.225·79-s − 4/9·81-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233308737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233308737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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
−8.679498853252053444462430241617, −8.655493509958517556104692087607, −8.378748037315465445192790829210, −7.980947296320422939728903060714, −7.58370080180617520189810021223, −6.52421319630927198344725128175, −5.95302503292927073396566116958, −5.47907564040983295563221596641, −5.25579521602379928421249093772, −4.29829876881027420494319425466, −3.66964895044272310843171490096, −3.61689003045514298236484607491, −2.92042524544419550067578696465, −1.88740825483186720696747488093, −0.68446346217697022883479913777,
0.68446346217697022883479913777, 1.88740825483186720696747488093, 2.92042524544419550067578696465, 3.61689003045514298236484607491, 3.66964895044272310843171490096, 4.29829876881027420494319425466, 5.25579521602379928421249093772, 5.47907564040983295563221596641, 5.95302503292927073396566116958, 6.52421319630927198344725128175, 7.58370080180617520189810021223, 7.980947296320422939728903060714, 8.378748037315465445192790829210, 8.655493509958517556104692087607, 8.679498853252053444462430241617
