L(s) = 1 | − 25·5-s + 108·7-s − 604·11-s − 306·13-s − 930·17-s + 1.32e3·19-s − 852·23-s + 625·25-s − 5.90e3·29-s + 3.32e3·31-s − 2.70e3·35-s + 1.07e4·37-s + 1.79e4·41-s − 9.26e3·43-s − 9.79e3·47-s − 5.14e3·49-s + 3.14e4·53-s + 1.51e4·55-s + 3.32e4·59-s − 4.02e4·61-s + 7.65e3·65-s − 5.88e4·67-s − 5.53e4·71-s + 2.72e4·73-s − 6.52e4·77-s − 3.14e4·79-s + 2.45e4·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.833·7-s − 1.50·11-s − 0.502·13-s − 0.780·17-s + 0.841·19-s − 0.335·23-s + 1/5·25-s − 1.30·29-s + 0.620·31-s − 0.372·35-s + 1.29·37-s + 1.66·41-s − 0.764·43-s − 0.646·47-s − 0.306·49-s + 1.53·53-s + 0.673·55-s + 1.24·59-s − 1.38·61-s + 0.224·65-s − 1.60·67-s − 1.30·71-s + 0.598·73-s − 1.25·77-s − 0.567·79-s + 0.391·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.499862423\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499862423\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
good | 7 | \( 1 - 108 T + p^{5} T^{2} \) |
| 11 | \( 1 + 604 T + p^{5} T^{2} \) |
| 13 | \( 1 + 306 T + p^{5} T^{2} \) |
| 17 | \( 1 + 930 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1324 T + p^{5} T^{2} \) |
| 23 | \( 1 + 852 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3320 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10774 T + p^{5} T^{2} \) |
| 41 | \( 1 - 438 p T + p^{5} T^{2} \) |
| 43 | \( 1 + 9264 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9796 T + p^{5} T^{2} \) |
| 53 | \( 1 - 31434 T + p^{5} T^{2} \) |
| 59 | \( 1 - 33228 T + p^{5} T^{2} \) |
| 61 | \( 1 + 40210 T + p^{5} T^{2} \) |
| 67 | \( 1 + 58864 T + p^{5} T^{2} \) |
| 71 | \( 1 + 55312 T + p^{5} T^{2} \) |
| 73 | \( 1 - 27258 T + p^{5} T^{2} \) |
| 79 | \( 1 + 31456 T + p^{5} T^{2} \) |
| 83 | \( 1 - 24552 T + p^{5} T^{2} \) |
| 89 | \( 1 - 90854 T + p^{5} T^{2} \) |
| 97 | \( 1 - 154706 T + p^{5} T^{2} \) |
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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
−9.694696565110888770148845964302, −8.671719551992234718174805449602, −7.73603621250066638226344309107, −7.42165932442220500106528821401, −5.97443235525483616496841408346, −5.04944915291965951486385486740, −4.32249007625561565064580150363, −2.96588333618455712978168990825, −2.00862005060660604072987308747, −0.55460790412269717911735885168,
0.55460790412269717911735885168, 2.00862005060660604072987308747, 2.96588333618455712978168990825, 4.32249007625561565064580150363, 5.04944915291965951486385486740, 5.97443235525483616496841408346, 7.42165932442220500106528821401, 7.73603621250066638226344309107, 8.671719551992234718174805449602, 9.694696565110888770148845964302