L(s) = 1 | − 3·5-s − 6·11-s − 5·13-s − 17-s + 2·19-s + 6·23-s + 4·25-s + 9·29-s − 7·31-s + 2·37-s + 9·41-s − 8·43-s − 3·47-s + 18·55-s − 3·59-s − 8·61-s + 15·65-s − 8·67-s + 12·71-s + 10·73-s + 16·79-s + 15·83-s + 3·85-s − 18·89-s − 6·95-s − 2·97-s + 101-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 1.80·11-s − 1.38·13-s − 0.242·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.25·31-s + 0.328·37-s + 1.40·41-s − 1.21·43-s − 0.437·47-s + 2.42·55-s − 0.390·59-s − 1.02·61-s + 1.86·65-s − 0.977·67-s + 1.42·71-s + 1.17·73-s + 1.80·79-s + 1.64·83-s + 0.325·85-s − 1.90·89-s − 0.615·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5566893834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5566893834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.62647733628390, −12.90123397315896, −12.50607751462060, −12.27369810369965, −11.64257195297803, −11.07280695639430, −10.75063422628665, −10.32653081214006, −9.511415509940853, −9.347471082373641, −8.356936421866860, −8.143634833653461, −7.612780364006764, −7.257892481671883, −6.806743800100602, −6.018535047845176, −5.211779376716356, −4.900908245347443, −4.592736560258587, −3.728817965195243, −3.134023116503916, −2.690231243897016, −2.139595208693322, −0.9978015343629200, −0.2639835635299397,
0.2639835635299397, 0.9978015343629200, 2.139595208693322, 2.690231243897016, 3.134023116503916, 3.728817965195243, 4.592736560258587, 4.900908245347443, 5.211779376716356, 6.018535047845176, 6.806743800100602, 7.257892481671883, 7.612780364006764, 8.143634833653461, 8.356936421866860, 9.347471082373641, 9.511415509940853, 10.32653081214006, 10.75063422628665, 11.07280695639430, 11.64257195297803, 12.27369810369965, 12.50607751462060, 12.90123397315896, 13.62647733628390