| L(s) = 1 | + 5.06·2-s + 3·3-s + 17.6·4-s − 5·5-s + 15.1·6-s − 27.4·7-s + 48.8·8-s + 9·9-s − 25.3·10-s + 52.9·12-s − 22.6·13-s − 138.·14-s − 15·15-s + 106.·16-s + 41.1·17-s + 45.5·18-s + 142.·19-s − 88.2·20-s − 82.3·21-s + 176.·23-s + 146.·24-s + 25·25-s − 114.·26-s + 27·27-s − 484.·28-s − 76.2·29-s − 75.9·30-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.20·4-s − 0.447·5-s + 1.03·6-s − 1.48·7-s + 2.16·8-s + 0.333·9-s − 0.800·10-s + 1.27·12-s − 0.484·13-s − 2.65·14-s − 0.258·15-s + 1.66·16-s + 0.587·17-s + 0.596·18-s + 1.71·19-s − 0.986·20-s − 0.855·21-s + 1.59·23-s + 1.24·24-s + 0.200·25-s − 0.867·26-s + 0.192·27-s − 3.26·28-s − 0.487·29-s − 0.462·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.573908424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.573908424\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.06T + 8T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 367.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 638.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 605.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 782.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 85.1T + 9.12e5T^{2} \) |
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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
−9.037236598205921855574468569671, −7.56538731061693665212905206945, −7.26307822358259254531316984862, −6.33738140739165496773939797422, −5.58428701515979438719512434920, −4.68295404203727403511285468873, −3.81745474533621915538725886623, −2.99298786691303386945408748121, −2.75192513907620275509136818853, −0.977004267892786146990232190982,
0.977004267892786146990232190982, 2.75192513907620275509136818853, 2.99298786691303386945408748121, 3.81745474533621915538725886623, 4.68295404203727403511285468873, 5.58428701515979438719512434920, 6.33738140739165496773939797422, 7.26307822358259254531316984862, 7.56538731061693665212905206945, 9.037236598205921855574468569671
