| L(s) = 1 | + 2·5-s − 2·11-s + 2·23-s + 3·25-s + 2·31-s + 2·37-s + 2·41-s + 2·43-s − 2·47-s − 4·55-s + 2·67-s − 2·71-s − 2·83-s − 2·97-s − 2·101-s + 2·103-s − 2·107-s + 4·115-s + 121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·11-s + 2·23-s + 3·25-s + 2·31-s + 2·37-s + 2·41-s + 2·43-s − 2·47-s − 4·55-s + 2·67-s − 2·71-s − 2·83-s − 2·97-s − 2·101-s + 2·103-s − 2·107-s + 4·115-s + 121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.200724629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200724629\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
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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
−9.073421811241552927831966085376, −8.738331015602951732028981966480, −8.271172739149102188118703992782, −7.86069572770182248889146000039, −7.65849991331762784159655355608, −7.07693253781228957577916211209, −6.70195060112868478299087943806, −6.35481563452777511878332141853, −5.93204099044015984028090742942, −5.64694940891030320170253981809, −5.13023926277635098893184360445, −5.11380009412380131379549430652, −4.32949296382578035542719194848, −4.31271459933775652156159640621, −3.08618063707521553158278071417, −2.85041318677267347682328820602, −2.49390221519112733229272636961, −2.38626684902158955179129635607, −1.25014256384810785344034605294, −1.07000138614464741192678716173,
1.07000138614464741192678716173, 1.25014256384810785344034605294, 2.38626684902158955179129635607, 2.49390221519112733229272636961, 2.85041318677267347682328820602, 3.08618063707521553158278071417, 4.31271459933775652156159640621, 4.32949296382578035542719194848, 5.11380009412380131379549430652, 5.13023926277635098893184360445, 5.64694940891030320170253981809, 5.93204099044015984028090742942, 6.35481563452777511878332141853, 6.70195060112868478299087943806, 7.07693253781228957577916211209, 7.65849991331762784159655355608, 7.86069572770182248889146000039, 8.271172739149102188118703992782, 8.738331015602951732028981966480, 9.073421811241552927831966085376
