L(s) = 1 | − 3-s + 9-s − 5·11-s − 2·13-s − 2·17-s − 6·19-s − 23-s − 27-s + 3·29-s − 4·31-s + 5·33-s − 5·37-s + 2·39-s + 4·41-s + 7·43-s + 10·47-s + 2·51-s + 2·53-s + 6·57-s − 10·59-s + 8·61-s − 7·67-s + 69-s + 3·71-s + 2·73-s + 11·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.485·17-s − 1.37·19-s − 0.208·23-s − 0.192·27-s + 0.557·29-s − 0.718·31-s + 0.870·33-s − 0.821·37-s + 0.320·39-s + 0.624·41-s + 1.06·43-s + 1.45·47-s + 0.280·51-s + 0.274·53-s + 0.794·57-s − 1.30·59-s + 1.02·61-s − 0.855·67-s + 0.120·69-s + 0.356·71-s + 0.234·73-s + 1.23·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.75832137650489, −13.93373614269783, −13.58102130926962, −12.95321151842072, −12.44287079554372, −12.32070253831529, −11.47304973914381, −10.85024355106505, −10.57124369612118, −10.25908870302675, −9.420952590338108, −8.981316833184409, −8.299004355987261, −7.788665358602236, −7.277761556395473, −6.708381681656203, −6.088781997749878, −5.523031108184598, −5.067241633349448, −4.397290006747318, −3.970488752189847, −2.962517722193113, −2.382035208888422, −1.871000091223573, −0.6866728649068853, 0,
0.6866728649068853, 1.871000091223573, 2.382035208888422, 2.962517722193113, 3.970488752189847, 4.397290006747318, 5.067241633349448, 5.523031108184598, 6.088781997749878, 6.708381681656203, 7.277761556395473, 7.788665358602236, 8.299004355987261, 8.981316833184409, 9.420952590338108, 10.25908870302675, 10.57124369612118, 10.85024355106505, 11.47304973914381, 12.32070253831529, 12.44287079554372, 12.95321151842072, 13.58102130926962, 13.93373614269783, 14.75832137650489