| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s + 5·7-s − 4·8-s + 3·9-s + 4·10-s − 6·12-s − 13-s − 10·14-s + 4·15-s + 5·16-s − 4·17-s − 6·18-s + 19-s − 6·20-s − 10·21-s + 3·23-s + 8·24-s + 3·25-s + 2·26-s − 4·27-s + 15·28-s − 6·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.88·7-s − 1.41·8-s + 9-s + 1.26·10-s − 1.73·12-s − 0.277·13-s − 2.67·14-s + 1.03·15-s + 5/4·16-s − 0.970·17-s − 1.41·18-s + 0.229·19-s − 1.34·20-s − 2.18·21-s + 0.625·23-s + 1.63·24-s + 3/5·25-s + 0.392·26-s − 0.769·27-s + 2.83·28-s − 1.11·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 37 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 95 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 27 T + 299 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 146 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 239 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.331420690523076946388381015927, −7.937240366909216501708192753363, −7.60911135715553597951826324073, −7.40789450789434373373568123206, −6.91650761147677290634420670057, −6.91007567223013795762542766240, −6.05238741391409862809572001622, −5.87427739772671129374641763921, −5.33701866257475819010616990689, −4.99524162532196862061768757021, −4.58828000927481859146914928855, −4.33419002470293939689708783656, −3.47334203154350304762918965027, −3.45271838883094074216869697218, −2.26626677514595035039078652646, −2.15819819502255734501431877720, −1.33913445469909408933744255824, −1.24016133863131232670551527694, 0, 0,
1.24016133863131232670551527694, 1.33913445469909408933744255824, 2.15819819502255734501431877720, 2.26626677514595035039078652646, 3.45271838883094074216869697218, 3.47334203154350304762918965027, 4.33419002470293939689708783656, 4.58828000927481859146914928855, 4.99524162532196862061768757021, 5.33701866257475819010616990689, 5.87427739772671129374641763921, 6.05238741391409862809572001622, 6.91007567223013795762542766240, 6.91650761147677290634420670057, 7.40789450789434373373568123206, 7.60911135715553597951826324073, 7.937240366909216501708192753363, 8.331420690523076946388381015927
