L(s) = 1 | − 2·3-s + 7-s + 9-s + 6·11-s + 4·13-s − 6·17-s − 8·19-s − 2·21-s + 23-s + 4·27-s + 6·29-s + 4·31-s − 12·33-s + 4·37-s − 8·39-s + 6·41-s − 10·43-s + 12·47-s + 49-s + 12·51-s + 16·57-s + 6·59-s + 14·61-s + 63-s + 14·67-s − 2·69-s − 14·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s − 0.436·21-s + 0.208·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 2.08·33-s + 0.657·37-s − 1.28·39-s + 0.937·41-s − 1.52·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 2.11·57-s + 0.781·59-s + 1.79·61-s + 0.125·63-s + 1.71·67-s − 0.240·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.956276200\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.956276200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.23250796192688, −13.71641399271573, −13.18469883217535, −12.58710877950345, −12.16860322602902, −11.48134851793285, −11.29295864679163, −10.93663097040838, −10.31010411039955, −9.763108960738974, −8.893527277767019, −8.537176433173033, −8.424846445364771, −7.189874064050129, −6.675059298530701, −6.345506055484596, −6.060752375622995, −5.270754391427104, −4.443990051474045, −4.287292683366526, −3.682463748880848, −2.608777373891628, −1.960327546365395, −1.102590764620942, −0.5992728275760118,
0.5992728275760118, 1.102590764620942, 1.960327546365395, 2.608777373891628, 3.682463748880848, 4.287292683366526, 4.443990051474045, 5.270754391427104, 6.060752375622995, 6.345506055484596, 6.675059298530701, 7.189874064050129, 8.424846445364771, 8.537176433173033, 8.893527277767019, 9.763108960738974, 10.31010411039955, 10.93663097040838, 11.29295864679163, 11.48134851793285, 12.16860322602902, 12.58710877950345, 13.18469883217535, 13.71641399271573, 14.23250796192688