| L(s) = 1 | − 5.61·2-s + 23.5·4-s − 87.4·8-s + 302.·16-s + 51.3·17-s + 61.8·19-s − 220.·23-s − 105.·31-s − 1.00e3·32-s − 288.·34-s − 347.·38-s + 1.23e3·46-s + 545.·47-s − 343·49-s + 85.1·53-s − 585.·61-s + 591.·62-s + 3.20e3·64-s + 1.20e3·68-s + 1.45e3·76-s + 1.03e3·79-s − 75.9·83-s − 5.18e3·92-s − 3.06e3·94-s + 1.92e3·98-s − 478.·106-s − 17.8·107-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 2.94·4-s − 3.86·8-s + 4.72·16-s + 0.732·17-s + 0.747·19-s − 1.99·23-s − 0.610·31-s − 5.53·32-s − 1.45·34-s − 1.48·38-s + 3.96·46-s + 1.69·47-s − 49-s + 0.220·53-s − 1.22·61-s + 1.21·62-s + 6.25·64-s + 2.15·68-s + 2.20·76-s + 1.47·79-s − 0.100·83-s − 5.87·92-s − 3.36·94-s + 1.98·98-s − 0.438·106-s − 0.0161·107-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 | \( 1 + 5.61T + 8T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 220.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 - 545.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 85.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 585.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 75.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.660839317980334728279658025978, −8.891506926970474656442660701946, −7.928474844757642243565290358620, −7.49498098405244483022029203739, −6.39631357039475826390192200533, −5.59741183224194471716479896832, −3.57156542603687301629924832523, −2.35637356037207912321532121796, −1.26835289995426341540392963987, 0,
1.26835289995426341540392963987, 2.35637356037207912321532121796, 3.57156542603687301629924832523, 5.59741183224194471716479896832, 6.39631357039475826390192200533, 7.49498098405244483022029203739, 7.928474844757642243565290358620, 8.891506926970474656442660701946, 9.660839317980334728279658025978
