| L(s) = 1 | − 12·7-s − 12·13-s − 16-s + 16·31-s − 12·37-s + 72·49-s − 40·61-s + 24·67-s + 24·73-s + 144·91-s + 48·97-s + 12·103-s + 12·112-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4.53·7-s − 3.32·13-s − 1/4·16-s + 2.87·31-s − 1.97·37-s + 72/7·49-s − 5.12·61-s + 2.93·67-s + 2.80·73-s + 15.0·91-s + 4.87·97-s + 1.18·103-s + 1.13·112-s + 8/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2572696333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2572696333\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 958 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.959558892980173253887597740478, −7.65359448086521705782576714688, −7.46730349907315239632715518476, −7.09271576405041976064577955761, −6.99885044727681020528064331254, −6.82827530481503565730328579584, −6.64009482320710821630093794064, −6.18633157034567024980305881875, −6.09543985722002574157990239534, −6.04357403697218999997145279834, −5.77398418277142679596444837232, −4.92358957330997213304595705975, −4.88743582269822032033618894449, −4.75782807574886707819103621006, −4.72851376793497843396238970036, −3.74708910966438985581309200602, −3.69561833850537807544706724742, −3.47921868070753582320687996485, −3.17072273292877850986959807072, −2.77578894851676729461070996670, −2.60039985594810480769743723767, −2.39994272450561436864078295071, −1.89124263769026721612858904367, −0.70700108465147644722161019265, −0.26062411282232008090013267252,
0.26062411282232008090013267252, 0.70700108465147644722161019265, 1.89124263769026721612858904367, 2.39994272450561436864078295071, 2.60039985594810480769743723767, 2.77578894851676729461070996670, 3.17072273292877850986959807072, 3.47921868070753582320687996485, 3.69561833850537807544706724742, 3.74708910966438985581309200602, 4.72851376793497843396238970036, 4.75782807574886707819103621006, 4.88743582269822032033618894449, 4.92358957330997213304595705975, 5.77398418277142679596444837232, 6.04357403697218999997145279834, 6.09543985722002574157990239534, 6.18633157034567024980305881875, 6.64009482320710821630093794064, 6.82827530481503565730328579584, 6.99885044727681020528064331254, 7.09271576405041976064577955761, 7.46730349907315239632715518476, 7.65359448086521705782576714688, 7.959558892980173253887597740478
