| L(s) = 1 | − 4.42·2-s + 11.6·4-s + 20.3·5-s − 15.9·8-s − 90.2·10-s − 70.0·11-s − 22.1·16-s + 236.·20-s + 310.·22-s + 290.·25-s + 225.·32-s − 325.·40-s + 486.·41-s − 452·43-s − 812.·44-s + 71.1·47-s − 343·49-s − 1.28e3·50-s − 1.42e3·55-s + 696.·59-s − 944.·61-s − 822.·64-s + 123.·71-s + 418.·79-s − 451.·80-s − 2.15e3·82-s − 1.50e3·83-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.45·4-s + 1.82·5-s − 0.705·8-s − 2.85·10-s − 1.91·11-s − 0.346·16-s + 2.64·20-s + 3.00·22-s + 2.32·25-s + 1.24·32-s − 1.28·40-s + 1.85·41-s − 1.60·43-s − 2.78·44-s + 0.220·47-s − 49-s − 3.64·50-s − 3.50·55-s + 1.53·59-s − 1.98·61-s − 1.60·64-s + 0.206·71-s + 0.595·79-s − 0.631·80-s − 2.90·82-s − 1.99·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.42T + 8T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 70.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 - 486.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 452T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 696.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 944.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 - 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 155.T + 7.04e5T^{2} \) |
| 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
−8.820524125677110181469314497388, −8.092123795331436383180762764707, −7.31494027396041057658711801639, −6.39890592584817335997313572899, −5.59841777465940521575435540804, −4.82528943994074521641636881511, −2.81889130521058540160829223361, −2.21194569838360631144232425907, −1.27193287679133555141862337288, 0,
1.27193287679133555141862337288, 2.21194569838360631144232425907, 2.81889130521058540160829223361, 4.82528943994074521641636881511, 5.59841777465940521575435540804, 6.39890592584817335997313572899, 7.31494027396041057658711801639, 8.092123795331436383180762764707, 8.820524125677110181469314497388
