L(s) = 1 | − 5-s − 11-s − 2·13-s − 3·17-s − 19-s + 6·23-s + 25-s + 9·29-s + 5·31-s + 5·37-s − 6·41-s − 8·43-s − 6·47-s − 9·53-s + 55-s − 6·59-s − 5·61-s + 2·65-s − 8·67-s − 9·71-s + 10·73-s − 14·79-s + 6·83-s + 3·85-s − 15·89-s + 95-s − 8·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.898·31-s + 0.821·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s − 1.23·53-s + 0.134·55-s − 0.781·59-s − 0.640·61-s + 0.248·65-s − 0.977·67-s − 1.06·71-s + 1.17·73-s − 1.57·79-s + 0.658·83-s + 0.325·85-s − 1.58·89-s + 0.102·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4743627412\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4743627412\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.41616884040640, −12.03423897298975, −11.56606660826043, −11.08524741603828, −10.75154082987435, −10.19660955455999, −9.764826614334999, −9.365477679664890, −8.674963290081247, −8.373294866484582, −8.017158621684334, −7.381085443626994, −6.928659483371182, −6.489031492840050, −6.168811389839278, −5.285803129826043, −4.924041293620427, −4.506596839169441, −4.129283797180829, −3.181877415774814, −2.938994434543885, −2.497064849904129, −1.566991899076654, −1.171199003144936, −0.1813437285826899,
0.1813437285826899, 1.171199003144936, 1.566991899076654, 2.497064849904129, 2.938994434543885, 3.181877415774814, 4.129283797180829, 4.506596839169441, 4.924041293620427, 5.285803129826043, 6.168811389839278, 6.489031492840050, 6.928659483371182, 7.381085443626994, 8.017158621684334, 8.373294866484582, 8.674963290081247, 9.365477679664890, 9.764826614334999, 10.19660955455999, 10.75154082987435, 11.08524741603828, 11.56606660826043, 12.03423897298975, 12.41616884040640