| L(s) = 1 | + 42.1·2-s − 81·3-s + 1.26e3·4-s + 625·5-s − 3.41e3·6-s + 4.30e3·7-s + 3.17e4·8-s + 6.56e3·9-s + 2.63e4·10-s + 7.42e4·11-s − 1.02e5·12-s + 1.62e5·13-s + 1.81e5·14-s − 5.06e4·15-s + 6.91e5·16-s + 3.54e5·17-s + 2.76e5·18-s − 1.30e5·19-s + 7.91e5·20-s − 3.48e5·21-s + 3.12e6·22-s − 2.24e6·23-s − 2.57e6·24-s + 3.90e5·25-s + 6.86e6·26-s − 5.31e5·27-s + 5.45e6·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s + 0.447·5-s − 1.07·6-s + 0.678·7-s + 2.74·8-s + 0.333·9-s + 0.833·10-s + 1.52·11-s − 1.42·12-s + 1.58·13-s + 1.26·14-s − 0.258·15-s + 2.63·16-s + 1.03·17-s + 0.621·18-s − 0.229·19-s + 1.10·20-s − 0.391·21-s + 2.84·22-s − 1.67·23-s − 1.58·24-s + 0.200·25-s + 2.94·26-s − 0.192·27-s + 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.958046627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.958046627\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 42.1T + 512T^{2} \) |
| 7 | \( 1 - 4.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.42e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.62e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.54e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 2.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.44e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.14e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.51e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.01e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.03e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.96e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.24e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.85e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.81e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.26e9T + 7.60e17T^{2} \) |
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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
−10.91567974195468930260311833470, −9.521530801394288290753516188399, −7.982311126817638678027758762992, −6.76035681095565787096460150944, −5.95258028396022398203676837081, −5.45770427771996681674743981004, −4.06242053349414453138764260961, −3.69156201432432069791864999326, −1.94148939953761022316329903198, −1.27056266863966110286936949603,
1.27056266863966110286936949603, 1.94148939953761022316329903198, 3.69156201432432069791864999326, 4.06242053349414453138764260961, 5.45770427771996681674743981004, 5.95258028396022398203676837081, 6.76035681095565787096460150944, 7.982311126817638678027758762992, 9.521530801394288290753516188399, 10.91567974195468930260311833470
