| L(s) = 1 | − 5·9-s − 2·13-s − 12·17-s − 10·25-s − 6·29-s + 16·37-s + 6·41-s − 10·49-s + 12·53-s + 28·61-s − 14·73-s + 16·81-s − 20·97-s + 12·101-s − 32·109-s + 36·113-s + 10·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 60·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 0.554·13-s − 2.91·17-s − 2·25-s − 1.11·29-s + 2.63·37-s + 0.937·41-s − 1.42·49-s + 1.64·53-s + 3.58·61-s − 1.63·73-s + 16/9·81-s − 2.03·97-s + 1.19·101-s − 3.06·109-s + 3.38·113-s + 0.924·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 4.85·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05292017083867367601940145437, −9.514891056616409629198104167660, −9.179275350265405931647106340341, −8.504802095603263578043187398864, −8.212075228103815323528954683241, −7.49853094467698855736649508236, −6.89620203002587049775804567261, −6.16371681572141052386909884198, −5.84967351690269779377701555372, −5.13331484784391238836917003859, −4.29263111870449453474910826205, −3.83402862781800864661702948677, −2.43652221488536246001833799986, −2.39273654521761793285798715610, 0,
2.39273654521761793285798715610, 2.43652221488536246001833799986, 3.83402862781800864661702948677, 4.29263111870449453474910826205, 5.13331484784391238836917003859, 5.84967351690269779377701555372, 6.16371681572141052386909884198, 6.89620203002587049775804567261, 7.49853094467698855736649508236, 8.212075228103815323528954683241, 8.504802095603263578043187398864, 9.179275350265405931647106340341, 9.514891056616409629198104167660, 10.05292017083867367601940145437
