L(s) = 1 | − 3-s + 9-s + 2·11-s − 2·13-s − 4·17-s − 4·19-s − 6·23-s − 5·25-s − 27-s + 2·29-s − 2·33-s + 6·37-s + 2·39-s − 8·41-s + 8·43-s + 4·47-s + 4·51-s + 6·53-s + 4·57-s − 14·61-s − 4·67-s + 6·69-s − 2·71-s + 2·73-s + 5·75-s + 4·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.986·37-s + 0.320·39-s − 1.24·41-s + 1.21·43-s + 0.583·47-s + 0.560·51-s + 0.824·53-s + 0.529·57-s − 1.79·61-s − 0.488·67-s + 0.722·69-s − 0.237·71-s + 0.234·73-s + 0.577·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.066245406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066245406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.60358930353858721893638903260, −6.98649547514778586458532404584, −6.09508828285552851184805426158, −5.99211098724561973112586840432, −4.77993360821194461421591970337, −4.34144512518339125071505160735, −3.62774855855583565758325147807, −2.41403782107225115278545940394, −1.79737977993024530874926914061, −0.49552899022940568638767842684,
0.49552899022940568638767842684, 1.79737977993024530874926914061, 2.41403782107225115278545940394, 3.62774855855583565758325147807, 4.34144512518339125071505160735, 4.77993360821194461421591970337, 5.99211098724561973112586840432, 6.09508828285552851184805426158, 6.98649547514778586458532404584, 7.60358930353858721893638903260