L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s + 6·11-s + 4·13-s + 2·15-s + 6·17-s − 8·19-s + 2·21-s + 23-s + 25-s − 4·27-s − 6·29-s − 4·31-s + 12·33-s + 35-s + 4·37-s + 8·39-s + 6·41-s + 10·43-s + 45-s + 12·47-s + 49-s + 12·51-s + 6·55-s − 16·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 0.516·15-s + 1.45·17-s − 1.83·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 2.08·33-s + 0.169·35-s + 0.657·37-s + 1.28·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.809·55-s − 2.11·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.013370386\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.013370386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.55377303271744, −14.09775572612511, −13.61004166573202, −13.09598634631409, −12.43249996533870, −12.12383817241780, −11.19420977864917, −10.98969976250739, −10.36597283469606, −9.486196311609667, −9.231338287251825, −8.809670818149547, −8.390604618137930, −7.618149022752519, −7.320109745823206, −6.383388583215708, −5.957102710212010, −5.588153787019808, −4.377116267274646, −4.014456465120180, −3.575585086436911, −2.812089345310708, −2.061181276825085, −1.533586821290975, −0.8413613461455647,
0.8413613461455647, 1.533586821290975, 2.061181276825085, 2.812089345310708, 3.575585086436911, 4.014456465120180, 4.377116267274646, 5.588153787019808, 5.957102710212010, 6.383388583215708, 7.320109745823206, 7.618149022752519, 8.390604618137930, 8.809670818149547, 9.231338287251825, 9.486196311609667, 10.36597283469606, 10.98969976250739, 11.19420977864917, 12.12383817241780, 12.43249996533870, 13.09598634631409, 13.61004166573202, 14.09775572612511, 14.55377303271744