| L(s) = 1 | − 3-s + 9-s − 8·11-s + 2·13-s + 6·25-s − 27-s + 8·33-s + 20·37-s − 2·39-s − 8·47-s − 10·49-s − 16·59-s − 28·61-s + 32·71-s − 20·73-s − 6·75-s + 81-s − 4·97-s − 8·99-s + 24·107-s − 4·109-s − 20·111-s + 2·117-s + 26·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 2.41·11-s + 0.554·13-s + 6/5·25-s − 0.192·27-s + 1.39·33-s + 3.28·37-s − 0.320·39-s − 1.16·47-s − 1.42·49-s − 2.08·59-s − 3.58·61-s + 3.79·71-s − 2.34·73-s − 0.692·75-s + 1/9·81-s − 0.406·97-s − 0.804·99-s + 2.32·107-s − 0.383·109-s − 1.89·111-s + 0.184·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−8.437984313060842025739365652378, −7.965645572263157998665195857010, −7.82743937659030627971434697795, −7.33786979345918290465240439010, −6.60471812809635507315319476307, −6.07506199316679251481919750358, −5.89249065254499898615542236081, −5.10290917264965015136465501487, −4.70084323749332305530012565841, −4.45984308910786813852795903758, −3.27214076659366118813611445645, −2.96541597463640709898029251011, −2.25446002723202452780259837823, −1.20578006642341497703303933331, 0,
1.20578006642341497703303933331, 2.25446002723202452780259837823, 2.96541597463640709898029251011, 3.27214076659366118813611445645, 4.45984308910786813852795903758, 4.70084323749332305530012565841, 5.10290917264965015136465501487, 5.89249065254499898615542236081, 6.07506199316679251481919750358, 6.60471812809635507315319476307, 7.33786979345918290465240439010, 7.82743937659030627971434697795, 7.965645572263157998665195857010, 8.437984313060842025739365652378
