| L(s) = 1 | + 0.873·3-s + 5-s + 3.64·7-s − 2.23·9-s − 1.06·13-s + 0.873·15-s + 0.252·17-s − 3.06·19-s + 3.18·21-s + 7.04·23-s + 25-s − 4.57·27-s + 1.61·29-s + 0.788·31-s + 3.64·35-s − 4.91·37-s − 0.932·39-s + 6.39·41-s + 2.63·43-s − 2.23·45-s + 7.78·47-s + 6.25·49-s + 0.220·51-s − 6.32·53-s − 2.67·57-s + 5.48·59-s − 6.80·61-s + ⋯ |
| L(s) = 1 | + 0.504·3-s + 0.447·5-s + 1.37·7-s − 0.745·9-s − 0.296·13-s + 0.225·15-s + 0.0613·17-s − 0.703·19-s + 0.693·21-s + 1.46·23-s + 0.200·25-s − 0.880·27-s + 0.299·29-s + 0.141·31-s + 0.615·35-s − 0.807·37-s − 0.149·39-s + 0.998·41-s + 0.401·43-s − 0.333·45-s + 1.13·47-s + 0.893·49-s + 0.0309·51-s − 0.868·53-s − 0.354·57-s + 0.713·59-s − 0.870·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.160236210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.160236210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.873T + 3T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 0.252T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 - 0.788T + 31T^{2} \) |
| 37 | \( 1 + 4.91T + 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 - 7.78T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 5.48T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 9.49T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 3.02T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 8.77T + 97T^{2} \) |
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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
−7.78382549899700784624703602499, −7.13174010560572989115756232627, −6.29294599684213836736239207056, −5.50935591198639786735026739901, −4.96869712831181508114689155441, −4.29332308087920611635555527005, −3.29775559006293402564425389741, −2.49344909383242381656060936434, −1.89019203855342015167072434932, −0.836795839869820155954220036456,
0.836795839869820155954220036456, 1.89019203855342015167072434932, 2.49344909383242381656060936434, 3.29775559006293402564425389741, 4.29332308087920611635555527005, 4.96869712831181508114689155441, 5.50935591198639786735026739901, 6.29294599684213836736239207056, 7.13174010560572989115756232627, 7.78382549899700784624703602499
