| L(s) = 1 | + 2·3-s + 5·4-s + 6·5-s + 6·7-s + 2·9-s + 10·12-s + 12·15-s + 12·16-s − 12·17-s + 30·20-s + 12·21-s + 5·25-s + 6·27-s + 30·28-s + 36·35-s + 10·36-s + 32·37-s − 8·41-s + 26·43-s + 12·45-s − 32·47-s + 24·48-s + 18·49-s − 24·51-s + 10·59-s + 60·60-s + 12·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 5/2·4-s + 2.68·5-s + 2.26·7-s + 2/3·9-s + 2.88·12-s + 3.09·15-s + 3·16-s − 2.91·17-s + 6.70·20-s + 2.61·21-s + 25-s + 1.15·27-s + 5.66·28-s + 6.08·35-s + 5/3·36-s + 5.26·37-s − 1.24·41-s + 3.96·43-s + 1.78·45-s − 4.66·47-s + 3.46·48-s + 18/7·49-s − 3.36·51-s + 1.30·59-s + 7.74·60-s + 1.51·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.13951187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.13951187\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 207 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 49 p T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2731 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 211 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 16 T + 133 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 81 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 15 T^{2} + 713 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 5 T + 23 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 209 T^{2} + 18081 T^{4} - 209 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 19 T + 223 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 149 T^{2} + 14101 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 210 T^{2} + 20963 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 253 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 5 T + 153 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7363 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.86770920276614059470928369557, −7.62623249791977015386424505931, −7.58738636717348328995607258045, −7.14608232646074804621520514852, −7.07449734467936911617856714257, −6.70668093817699346898275281593, −6.32140639643068285653828306227, −6.07842113778678663209393091182, −6.06400760783670640779205496331, −5.88950177272823048780742314086, −5.88171342667040902915940334340, −5.04290162688712199285765151897, −4.84315322848213187018152379581, −4.82728574092827190671717886850, −4.17944931062128854471464480788, −4.17227825748126203774238747866, −3.92329811350257781320650975924, −2.74318620388158491255135281557, −2.73608755653770352116939033284, −2.68950323975525126104993337524, −2.57099904858599971441612436658, −1.98694529128153579213550212957, −1.67831645586395942662184152274, −1.63055850491832581965360534497, −1.39647026846428147050162044608,
1.39647026846428147050162044608, 1.63055850491832581965360534497, 1.67831645586395942662184152274, 1.98694529128153579213550212957, 2.57099904858599971441612436658, 2.68950323975525126104993337524, 2.73608755653770352116939033284, 2.74318620388158491255135281557, 3.92329811350257781320650975924, 4.17227825748126203774238747866, 4.17944931062128854471464480788, 4.82728574092827190671717886850, 4.84315322848213187018152379581, 5.04290162688712199285765151897, 5.88171342667040902915940334340, 5.88950177272823048780742314086, 6.06400760783670640779205496331, 6.07842113778678663209393091182, 6.32140639643068285653828306227, 6.70668093817699346898275281593, 7.07449734467936911617856714257, 7.14608232646074804621520514852, 7.58738636717348328995607258045, 7.62623249791977015386424505931, 7.86770920276614059470928369557
