| L(s) = 1 | + 2·5-s − 4·7-s + 4·13-s − 17-s − 8·19-s − 25-s − 10·31-s − 8·35-s + 8·37-s − 10·41-s − 8·43-s + 10·47-s + 9·49-s + 12·53-s − 8·59-s + 2·61-s + 8·65-s − 4·67-s − 4·71-s − 10·73-s + 12·79-s − 4·83-s − 2·85-s + 6·89-s − 16·91-s − 16·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 1.10·13-s − 0.242·17-s − 1.83·19-s − 1/5·25-s − 1.79·31-s − 1.35·35-s + 1.31·37-s − 1.56·41-s − 1.21·43-s + 1.45·47-s + 9/7·49-s + 1.64·53-s − 1.04·59-s + 0.256·61-s + 0.992·65-s − 0.488·67-s − 0.474·71-s − 1.17·73-s + 1.35·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s − 1.67·91-s − 1.64·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6065921455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6065921455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.87105499411334, −12.29177329310425, −11.94772882020782, −11.12868723410707, −10.81789439274362, −10.38073140316180, −9.938420479078663, −9.526255671874009, −8.969703239704810, −8.720505320191827, −8.242687956697099, −7.415333761308687, −6.991190922583306, −6.400030809179299, −6.202250986224646, −5.763092438328291, −5.283013695832956, −4.460022504451482, −3.893788951373637, −3.602022004477286, −2.906147396189305, −2.324007574075581, −1.852209225082719, −1.185215468663052, −0.2039938008768387,
0.2039938008768387, 1.185215468663052, 1.852209225082719, 2.324007574075581, 2.906147396189305, 3.602022004477286, 3.893788951373637, 4.460022504451482, 5.283013695832956, 5.763092438328291, 6.202250986224646, 6.400030809179299, 6.991190922583306, 7.415333761308687, 8.242687956697099, 8.720505320191827, 8.969703239704810, 9.526255671874009, 9.938420479078663, 10.38073140316180, 10.81789439274362, 11.12868723410707, 11.94772882020782, 12.29177329310425, 12.87105499411334
