L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 13-s + 15-s − 6·17-s − 4·19-s + 4·21-s + 8·23-s + 25-s + 27-s + 4·35-s + 12·37-s − 39-s + 10·41-s + 4·43-s + 45-s + 6·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s − 4·59-s + 6·61-s + 4·63-s − 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.676·35-s + 1.97·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.682422122\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.682422122\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.27184680933690, −12.69113522459114, −12.32906233411092, −11.54337215637138, −11.11056975793526, −10.82484112701081, −10.52360064227738, −9.581617343968429, −9.192163145821601, −8.985183992910900, −8.267737934322789, −7.958469701693121, −7.480831675459749, −6.770907995076066, −6.530451797878380, −5.706603794789206, −5.206071796637991, −4.673770463369687, −4.249591338960918, −3.852372218900036, −2.697268107324562, −2.469222804003325, −2.029520812675119, −1.192353297643562, −0.6986044757992831,
0.6986044757992831, 1.192353297643562, 2.029520812675119, 2.469222804003325, 2.697268107324562, 3.852372218900036, 4.249591338960918, 4.673770463369687, 5.206071796637991, 5.706603794789206, 6.530451797878380, 6.770907995076066, 7.480831675459749, 7.958469701693121, 8.267737934322789, 8.985183992910900, 9.192163145821601, 9.581617343968429, 10.52360064227738, 10.82484112701081, 11.11056975793526, 11.54337215637138, 12.32906233411092, 12.69113522459114, 13.27184680933690