L(s) = 1 | − 3·3-s − 13·7-s + 9·9-s + 6·11-s + 5·13-s − 78·17-s + 65·19-s + 39·21-s + 138·23-s − 27·27-s + 66·29-s + 299·31-s − 18·33-s − 214·37-s − 15·39-s + 360·41-s + 203·43-s + 78·47-s − 174·49-s + 234·51-s + 636·53-s − 195·57-s + 786·59-s + 467·61-s − 117·63-s − 217·67-s − 414·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.701·7-s + 1/3·9-s + 0.164·11-s + 0.106·13-s − 1.11·17-s + 0.784·19-s + 0.405·21-s + 1.25·23-s − 0.192·27-s + 0.422·29-s + 1.73·31-s − 0.0949·33-s − 0.950·37-s − 0.0615·39-s + 1.37·41-s + 0.719·43-s + 0.242·47-s − 0.507·49-s + 0.642·51-s + 1.64·53-s − 0.453·57-s + 1.73·59-s + 0.980·61-s − 0.233·63-s − 0.395·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.324359433\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.324359433\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 65 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 299 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 203 T + p^{3} T^{2} \) |
| 47 | \( 1 - 78 T + p^{3} T^{2} \) |
| 53 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 786 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 + 217 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 286 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 511 T + p^{3} T^{2} \) |
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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
−11.33084464678479667356745446546, −10.42744002071793728210232350532, −9.500193291056395887631122352923, −8.568511882645077864369180582756, −7.16143638687420049851341504230, −6.45332309417483332528698938193, −5.32132727864424098005152775626, −4.15018764099051957183511425453, −2.72466153288255976674702044996, −0.832408142692456753905514203754,
0.832408142692456753905514203754, 2.72466153288255976674702044996, 4.15018764099051957183511425453, 5.32132727864424098005152775626, 6.45332309417483332528698938193, 7.16143638687420049851341504230, 8.568511882645077864369180582756, 9.500193291056395887631122352923, 10.42744002071793728210232350532, 11.33084464678479667356745446546