| L(s) = 1 | − 0.258·2-s − 7.93·4-s − 14.5·7-s + 4.12·8-s + 49.2·11-s − 72.1·13-s + 3.75·14-s + 62.3·16-s + 118.·17-s + 123.·19-s − 12.7·22-s − 91.4·23-s + 18.6·26-s + 115.·28-s − 174.·29-s − 46.2·31-s − 49.1·32-s − 30.5·34-s − 154.·37-s − 31.9·38-s + 364.·41-s − 125.·43-s − 390.·44-s + 23.6·46-s − 221.·47-s − 132.·49-s + 572.·52-s + ⋯ |
| L(s) = 1 | − 0.0914·2-s − 0.991·4-s − 0.783·7-s + 0.182·8-s + 1.35·11-s − 1.53·13-s + 0.0716·14-s + 0.974·16-s + 1.68·17-s + 1.48·19-s − 0.123·22-s − 0.829·23-s + 0.140·26-s + 0.777·28-s − 1.11·29-s − 0.268·31-s − 0.271·32-s − 0.154·34-s − 0.688·37-s − 0.136·38-s + 1.38·41-s − 0.445·43-s − 1.33·44-s + 0.0758·46-s − 0.687·47-s − 0.385·49-s + 1.52·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.258T + 8T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 - 49.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 13.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 335.T + 9.12e5T^{2} \) |
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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
−9.735319852133442835443044694697, −9.106309948557660823742550032545, −7.84571209656769414474315739518, −7.20308636640991476464467929224, −5.91313362681760704864163283228, −5.12452559077187558901512581917, −3.94257719189974779877219500002, −3.15231960446786648172493864248, −1.32324874297173100205937929991, 0,
1.32324874297173100205937929991, 3.15231960446786648172493864248, 3.94257719189974779877219500002, 5.12452559077187558901512581917, 5.91313362681760704864163283228, 7.20308636640991476464467929224, 7.84571209656769414474315739518, 9.106309948557660823742550032545, 9.735319852133442835443044694697
