L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 2·11-s + 15-s − 2·17-s − 19-s + 2·21-s + 8·23-s + 25-s + 27-s + 2·33-s + 2·35-s + 4·37-s − 8·41-s + 6·43-s + 45-s + 8·47-s − 3·49-s − 2·51-s − 10·53-s + 2·55-s − 57-s + 8·59-s + 2·61-s + 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.348·33-s + 0.338·35-s + 0.657·37-s − 1.24·41-s + 0.914·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 0.269·55-s − 0.132·57-s + 1.04·59-s + 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.207901040\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.207901040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.385745069815320854220957068131, −7.67196202281110987888905109705, −6.89380098463423095036730475452, −6.27725956748470305833624123732, −5.23232181926895638499259159068, −4.64007537297163469773012860491, −3.76496173577229921926704089711, −2.81192603270148126756524350136, −1.95430284217611176639498989066, −1.04284208304124357235556805364,
1.04284208304124357235556805364, 1.95430284217611176639498989066, 2.81192603270148126756524350136, 3.76496173577229921926704089711, 4.64007537297163469773012860491, 5.23232181926895638499259159068, 6.27725956748470305833624123732, 6.89380098463423095036730475452, 7.67196202281110987888905109705, 8.385745069815320854220957068131