| L(s) = 1 | + 2·3-s − 4-s − 5-s − 7-s + 3·9-s + 11-s − 2·12-s + 13-s − 2·15-s − 2·17-s + 20-s − 2·21-s + 4·27-s + 28-s − 2·29-s + 2·33-s + 35-s − 3·36-s + 2·39-s − 44-s − 3·45-s + 47-s − 4·51-s − 52-s − 55-s + 2·60-s − 3·63-s + ⋯ |
| L(s) = 1 | + 2·3-s − 4-s − 5-s − 7-s + 3·9-s + 11-s − 2·12-s + 13-s − 2·15-s − 2·17-s + 20-s − 2·21-s + 4·27-s + 28-s − 2·29-s + 2·33-s + 35-s − 3·36-s + 2·39-s − 44-s − 3·45-s + 47-s − 4·51-s − 52-s − 55-s + 2·60-s − 3·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7703627276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7703627276\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−12.14934561063154958854333425236, −11.83881925072780939452790170688, −10.97901009140078801521546686572, −10.80856202486761247304944346622, −9.977854288867507135827628902561, −9.456569191986250616637180472093, −9.170630226804899325818910105422, −8.889002196645706859135933612168, −8.574958860136773651939128232757, −7.998381386217842099469961426020, −7.44906671391731656241861529965, −6.94005287664058914355889217412, −6.56583961080603925279781164907, −5.79368019381748874088411136543, −4.57848228841149103640878300130, −4.28595429585172042586611837020, −3.76475874034298914605932328799, −3.53913634701750426091528981135, −2.63303018824714148227077228516, −1.70821105391874785835466218405,
1.70821105391874785835466218405, 2.63303018824714148227077228516, 3.53913634701750426091528981135, 3.76475874034298914605932328799, 4.28595429585172042586611837020, 4.57848228841149103640878300130, 5.79368019381748874088411136543, 6.56583961080603925279781164907, 6.94005287664058914355889217412, 7.44906671391731656241861529965, 7.998381386217842099469961426020, 8.574958860136773651939128232757, 8.889002196645706859135933612168, 9.170630226804899325818910105422, 9.456569191986250616637180472093, 9.977854288867507135827628902561, 10.80856202486761247304944346622, 10.97901009140078801521546686572, 11.83881925072780939452790170688, 12.14934561063154958854333425236
