| L(s) = 1 | + 3-s + 5-s + 2·7-s − 9-s − 5·11-s + 15-s − 8·17-s + 19-s + 2·21-s + 10·23-s − 5·25-s + 7·29-s − 4·31-s − 5·33-s + 2·35-s − 15·37-s + 13·41-s − 3·43-s − 45-s − 6·47-s + 3·49-s − 8·51-s − 5·53-s − 5·55-s + 57-s − 2·59-s − 15·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 1/3·9-s − 1.50·11-s + 0.258·15-s − 1.94·17-s + 0.229·19-s + 0.436·21-s + 2.08·23-s − 25-s + 1.29·29-s − 0.718·31-s − 0.870·33-s + 0.338·35-s − 2.46·37-s + 2.03·41-s − 0.457·43-s − 0.149·45-s − 0.875·47-s + 3/7·49-s − 1.12·51-s − 0.686·53-s − 0.674·55-s + 0.132·57-s − 0.260·59-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89567296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89567296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 126 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 120 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 140 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 177 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 174 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 284 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.55445225196795659506014540856, −7.11211882275880355172690670590, −7.07969879258553135315993083467, −6.52025552238591896473779359709, −6.00103729285915785603211722937, −5.99797213808574791708517799816, −5.35061786851363515566173510883, −4.97727626217686690528396621333, −4.80735761838766296915394117107, −4.62512828043764854895244787680, −4.04087561822475568284476404701, −3.47833930757049119101987899811, −3.14384790445618922972432740739, −2.77086151216236492565209165535, −2.36028312192662227230132455739, −2.17343964266523719671760692213, −1.47079979878423314061268924014, −1.20500873092441719184971035453, 0, 0,
1.20500873092441719184971035453, 1.47079979878423314061268924014, 2.17343964266523719671760692213, 2.36028312192662227230132455739, 2.77086151216236492565209165535, 3.14384790445618922972432740739, 3.47833930757049119101987899811, 4.04087561822475568284476404701, 4.62512828043764854895244787680, 4.80735761838766296915394117107, 4.97727626217686690528396621333, 5.35061786851363515566173510883, 5.99797213808574791708517799816, 6.00103729285915785603211722937, 6.52025552238591896473779359709, 7.07969879258553135315993083467, 7.11211882275880355172690670590, 7.55445225196795659506014540856
