L(s) = 1 | − 3-s + 4·5-s + 7-s + 9-s + 2·11-s + 4·13-s − 4·15-s + 19-s − 21-s + 6·23-s + 11·25-s − 27-s + 10·29-s − 2·33-s + 4·35-s + 6·37-s − 4·39-s − 10·41-s − 8·43-s + 4·45-s − 12·47-s + 49-s − 6·53-s + 8·55-s − 57-s + 12·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 1.03·15-s + 0.229·19-s − 0.218·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s − 0.348·33-s + 0.676·35-s + 0.986·37-s − 0.640·39-s − 1.56·41-s − 1.21·43-s + 0.596·45-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.132·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.143396830\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.143396830\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.266803391345024683877420731014, −6.90517110955387543156167355371, −6.52915646399658381764957849000, −6.02007527522374924092131184302, −5.11579597036161923704227864003, −4.81620945922293785168875731596, −3.55217946131977538967608243251, −2.65494041436671545198249735692, −1.55539045038415570332568801933, −1.10890742239055138080490634639,
1.10890742239055138080490634639, 1.55539045038415570332568801933, 2.65494041436671545198249735692, 3.55217946131977538967608243251, 4.81620945922293785168875731596, 5.11579597036161923704227864003, 6.02007527522374924092131184302, 6.52915646399658381764957849000, 6.90517110955387543156167355371, 8.266803391345024683877420731014