| L(s) = 1 | − 2·5-s + 2·7-s + 8·17-s + 8·19-s + 10·23-s + 25-s + 6·29-s + 12·31-s − 4·35-s − 24·37-s − 10·41-s + 4·43-s + 14·47-s + 9·49-s + 8·53-s + 12·59-s + 6·61-s − 2·67-s + 16·73-s − 4·79-s + 6·83-s − 16·85-s + 28·89-s − 16·95-s − 4·97-s − 4·101-s + 20·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.94·17-s + 1.83·19-s + 2.08·23-s + 1/5·25-s + 1.11·29-s + 2.15·31-s − 0.676·35-s − 3.94·37-s − 1.56·41-s + 0.609·43-s + 2.04·47-s + 9/7·49-s + 1.09·53-s + 1.56·59-s + 0.768·61-s − 0.244·67-s + 1.87·73-s − 0.450·79-s + 0.658·83-s − 1.73·85-s + 2.96·89-s − 1.64·95-s − 0.406·97-s − 0.398·101-s + 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.451510079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.451510079\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T + 35 T^{2} - 190 T^{3} + 1396 T^{4} - 190 p T^{5} + 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 90 T^{3} - 36 T^{4} + 90 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 70 T^{2} - 144 T^{3} + 51 T^{4} - 144 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 10 T + 17 T^{2} + 10 T^{3} + 1108 T^{4} + 10 p T^{5} + 17 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 368 T^{3} - 2501 T^{4} + 368 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 14 T + 59 T^{2} - 602 T^{3} + 7348 T^{4} - 602 p T^{5} + 59 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 14 T^{2} - 144 T^{3} + 4923 T^{4} - 144 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 19 T^{2} - 298 T^{3} - 4532 T^{4} - 298 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 122 T^{2} - 80 T^{3} + 11539 T^{4} - 80 p T^{5} - 122 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 14 T + 131 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.75995113304210617087790432692, −6.01018507917860200364504927600, −6.00871199429714224143126170022, −5.91736405618493464233945516111, −5.38295945371299781371473888417, −5.24118409537129901167083672276, −5.18278829129028159646251404451, −5.15058730198881321357027395488, −4.91083687527932679981700741628, −4.50241615920619079438972761730, −4.35414153523709184134041458416, −4.00146965476361049211770660167, −3.89550821779315251172974022675, −3.35803792884046277530537631238, −3.35257568424756317001976607024, −3.20358041404664131716185968232, −3.20147267838816799584546743316, −2.54460583341413070962077348698, −2.38361779170187557576398274131, −2.12651015412862400858851335802, −1.72929136274523278830976506216, −1.22612582279908290900198994644, −1.04409572270346864466305798018, −0.77632012352639139228655465572, −0.61220779306442116758033154828,
0.61220779306442116758033154828, 0.77632012352639139228655465572, 1.04409572270346864466305798018, 1.22612582279908290900198994644, 1.72929136274523278830976506216, 2.12651015412862400858851335802, 2.38361779170187557576398274131, 2.54460583341413070962077348698, 3.20147267838816799584546743316, 3.20358041404664131716185968232, 3.35257568424756317001976607024, 3.35803792884046277530537631238, 3.89550821779315251172974022675, 4.00146965476361049211770660167, 4.35414153523709184134041458416, 4.50241615920619079438972761730, 4.91083687527932679981700741628, 5.15058730198881321357027395488, 5.18278829129028159646251404451, 5.24118409537129901167083672276, 5.38295945371299781371473888417, 5.91736405618493464233945516111, 6.00871199429714224143126170022, 6.01018507917860200364504927600, 6.75995113304210617087790432692
