| L(s) = 1 | − 3-s + 9-s − 2·11-s − 4·13-s − 6·25-s − 27-s + 2·33-s + 20·37-s + 4·39-s + 16·47-s − 10·49-s − 24·59-s + 20·61-s − 16·71-s + 12·73-s + 6·75-s + 81-s − 32·83-s − 4·97-s − 2·99-s − 20·109-s − 20·111-s − 4·117-s + 3·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 6/5·25-s − 0.192·27-s + 0.348·33-s + 3.28·37-s + 0.640·39-s + 2.33·47-s − 1.42·49-s − 3.12·59-s + 2.56·61-s − 1.89·71-s + 1.40·73-s + 0.692·75-s + 1/9·81-s − 3.51·83-s − 0.406·97-s − 0.201·99-s − 1.91·109-s − 1.89·111-s − 0.369·117-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.932199088171823379479580947914, −8.071554028904627902844738284977, −7.956659964685635211800714779949, −7.34087378011120859262644430825, −7.05430755180880074285547369845, −6.20097915233518496322100006686, −5.92948814461081293712003874605, −5.44620559411548300463639810762, −4.75118710346056158176317605927, −4.37492205611691939780872013052, −3.79814948526845934439056144160, −2.71552905653600617964922283600, −2.46483616139600209242493840583, −1.29948084140706303387979501539, 0,
1.29948084140706303387979501539, 2.46483616139600209242493840583, 2.71552905653600617964922283600, 3.79814948526845934439056144160, 4.37492205611691939780872013052, 4.75118710346056158176317605927, 5.44620559411548300463639810762, 5.92948814461081293712003874605, 6.20097915233518496322100006686, 7.05430755180880074285547369845, 7.34087378011120859262644430825, 7.956659964685635211800714779949, 8.071554028904627902844738284977, 8.932199088171823379479580947914
