L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 4·11-s − 2·13-s − 15-s + 19-s − 2·21-s + 8·23-s + 25-s + 27-s − 2·29-s + 10·31-s − 4·33-s + 2·35-s + 6·37-s − 2·39-s + 12·41-s + 8·43-s − 45-s + 8·47-s − 3·49-s − 6·53-s + 4·55-s + 57-s − 6·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.229·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.696·33-s + 0.338·35-s + 0.986·37-s − 0.320·39-s + 1.87·41-s + 1.21·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s + 0.132·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638318892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638318892\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.147419761271871272354642937619, −8.094892290976825000288656742302, −7.60454808441980285120122662578, −6.87026565439754025500762199807, −5.89915736099352673808280918566, −4.92416488810909051901283951074, −4.14212641887134945016435736327, −2.92936483576798041344267272221, −2.61725081168138791694652154838, −0.792182304489764548174574988163,
0.792182304489764548174574988163, 2.61725081168138791694652154838, 2.92936483576798041344267272221, 4.14212641887134945016435736327, 4.92416488810909051901283951074, 5.89915736099352673808280918566, 6.87026565439754025500762199807, 7.60454808441980285120122662578, 8.094892290976825000288656742302, 9.147419761271871272354642937619