| L(s) = 1 | − 16·2-s − 76.4·3-s + 256·4-s − 2.44e3·5-s + 1.22e3·6-s + 1.78e3·7-s − 4.09e3·8-s − 1.38e4·9-s + 3.90e4·10-s − 5.50e4·11-s − 1.95e4·12-s + 2.85e4·13-s − 2.86e4·14-s + 1.86e5·15-s + 6.55e4·16-s + 6.37e5·17-s + 2.21e5·18-s + 8.48e5·19-s − 6.24e5·20-s − 1.36e5·21-s + 8.80e5·22-s − 2.30e6·23-s + 3.13e5·24-s + 4.00e6·25-s − 4.56e5·26-s + 2.56e6·27-s + 4.57e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.545·3-s + 0.5·4-s − 1.74·5-s + 0.385·6-s + 0.281·7-s − 0.353·8-s − 0.702·9-s + 1.23·10-s − 1.13·11-s − 0.272·12-s + 0.277·13-s − 0.199·14-s + 0.952·15-s + 0.250·16-s + 1.85·17-s + 0.496·18-s + 1.49·19-s − 0.873·20-s − 0.153·21-s + 0.801·22-s − 1.72·23-s + 0.192·24-s + 2.05·25-s − 0.196·26-s + 0.928·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5148128403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5148128403\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 76.4T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.44e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.78e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.50e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 6.37e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.30e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.39e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 8.53e5T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.40e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.36e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.03e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.87e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.27e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.65e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.61e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.93e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−15.88580307857855730363665728436, −14.40634618404386616612844414466, −12.15838948731671798459949315541, −11.56169234923557941638289628921, −10.30389553703651258467397088437, −8.252560588214758011598271335686, −7.55533746520910779436546792741, −5.42741914896907171850221760046, −3.33317306650557759935701915811, −0.60148432085787545058140478930,
0.60148432085787545058140478930, 3.33317306650557759935701915811, 5.42741914896907171850221760046, 7.55533746520910779436546792741, 8.252560588214758011598271335686, 10.30389553703651258467397088437, 11.56169234923557941638289628921, 12.15838948731671798459949315541, 14.40634618404386616612844414466, 15.88580307857855730363665728436
