| L(s) = 1 | + 2-s + 6·3-s − 5·4-s + 10·5-s + 6·6-s + 7-s − 3·8-s + 27·9-s + 10·10-s − 30·12-s + 47·13-s + 14-s + 60·15-s − 29·16-s − 98·17-s + 27·18-s − 75·19-s − 50·20-s + 6·21-s − 57·23-s − 18·24-s + 75·25-s + 47·26-s + 108·27-s − 5·28-s − 127·29-s + 60·30-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1.15·3-s − 5/8·4-s + 0.894·5-s + 0.408·6-s + 0.0539·7-s − 0.132·8-s + 9-s + 0.316·10-s − 0.721·12-s + 1.00·13-s + 0.0190·14-s + 1.03·15-s − 0.453·16-s − 1.39·17-s + 0.353·18-s − 0.905·19-s − 0.559·20-s + 0.0623·21-s − 0.516·23-s − 0.153·24-s + 3/5·25-s + 0.354·26-s + 0.769·27-s − 0.0337·28-s − 0.813·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 594 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 47 T + 4936 T^{2} - 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 98 T + 12063 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 75 T + 14294 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 57 T + 15296 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 127 T + 52308 T^{2} + 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 183 T + 63434 T^{2} + 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 229 T + 50446 T^{2} + 229 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 18 T + 137554 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 186 T + 137774 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 615 T + 258896 T^{2} + 615 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 927 T + 512576 T^{2} + 927 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 380 T + 129354 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 509 T + 489940 T^{2} - 509 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1138 T + 694662 T^{2} + 1138 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 273 T + 574298 T^{2} + 273 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 440 T + 815938 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 973 T + 1173960 T^{2} - 973 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 864574 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1288 T + 1792530 T^{2} + 1288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 76 T + 1799074 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.610531212247136331178139926527, −8.598589875853202426457129744490, −8.035122545384632581259806876668, −7.60242878441684910899767958349, −7.16812094674502770078198575038, −6.53891211914455939643634306485, −6.34489166767670248644275643056, −6.09918410394423305947878629641, −5.16260555057876345865275581251, −5.11954280602755234543018670329, −4.41224792007397389748013505921, −4.24025502974766166786762098241, −3.58148254422038250591354585898, −3.33119060220014748184995613980, −2.68098720031604988372088964112, −2.01314547814520529775825780244, −1.78958960317689192300296613911, −1.36718175397376415773211921976, 0, 0,
1.36718175397376415773211921976, 1.78958960317689192300296613911, 2.01314547814520529775825780244, 2.68098720031604988372088964112, 3.33119060220014748184995613980, 3.58148254422038250591354585898, 4.24025502974766166786762098241, 4.41224792007397389748013505921, 5.11954280602755234543018670329, 5.16260555057876345865275581251, 6.09918410394423305947878629641, 6.34489166767670248644275643056, 6.53891211914455939643634306485, 7.16812094674502770078198575038, 7.60242878441684910899767958349, 8.035122545384632581259806876668, 8.598589875853202426457129744490, 8.610531212247136331178139926527
