| L(s) = 1 | + 4·3-s + 4-s + 4·5-s + 6·9-s + 4·12-s + 16·15-s + 16-s + 4·20-s + 16·23-s + 2·25-s − 4·27-s + 6·36-s + 24·45-s − 16·47-s + 4·48-s − 14·49-s + 12·53-s + 20·59-s + 16·60-s + 64-s − 16·67-s + 64·69-s + 16·71-s + 8·75-s + 4·80-s − 37·81-s + 2·89-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1/2·4-s + 1.78·5-s + 2·9-s + 1.15·12-s + 4.13·15-s + 1/4·16-s + 0.894·20-s + 3.33·23-s + 2/5·25-s − 0.769·27-s + 36-s + 3.57·45-s − 2.33·47-s + 0.577·48-s − 2·49-s + 1.64·53-s + 2.60·59-s + 2.06·60-s + 1/8·64-s − 1.95·67-s + 7.70·69-s + 1.89·71-s + 0.923·75-s + 0.447·80-s − 4.11·81-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.16646121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.16646121\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−7.60544094667500853617510180035, −7.08662686698941195360404919360, −6.67854632460910561231510247828, −6.31642557119714749591460530685, −5.82749657516878895125232295673, −5.19017598972715220168949521528, −5.18755879988201558728632555274, −4.41965058662194829080513185467, −3.66215748399741404907946534759, −3.16476680941726517752422559781, −3.11926741866275439679505800539, −2.48900639559223393465645037103, −2.01056152480496856645260763012, −1.83115187730622718981861867906, −1.00021285205620777511821333269,
1.00021285205620777511821333269, 1.83115187730622718981861867906, 2.01056152480496856645260763012, 2.48900639559223393465645037103, 3.11926741866275439679505800539, 3.16476680941726517752422559781, 3.66215748399741404907946534759, 4.41965058662194829080513185467, 5.18755879988201558728632555274, 5.19017598972715220168949521528, 5.82749657516878895125232295673, 6.31642557119714749591460530685, 6.67854632460910561231510247828, 7.08662686698941195360404919360, 7.60544094667500853617510180035
