| L(s) = 1 | + 2-s + 4-s + 3·5-s + 3·8-s + 3·10-s + 11-s − 5·13-s + 16-s + 2·17-s − 3·19-s + 3·20-s + 22-s + 9·23-s + 25-s − 5·26-s + 2·31-s − 32-s + 2·34-s − 2·37-s − 3·38-s + 9·40-s − 3·41-s − 3·43-s + 44-s + 9·46-s − 14·47-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.06·8-s + 0.948·10-s + 0.301·11-s − 1.38·13-s + 1/4·16-s + 0.485·17-s − 0.688·19-s + 0.670·20-s + 0.213·22-s + 1.87·23-s + 1/5·25-s − 0.980·26-s + 0.359·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.486·38-s + 1.42·40-s − 0.468·41-s − 0.457·43-s + 0.150·44-s + 1.32·46-s − 2.04·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56205009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56205009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.499210970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.499210970\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.017216452473982332983097595015, −7.54481535332188036567132817036, −7.30636815171210682285208297141, −6.81927287636335356788331689782, −6.75093309700779139201493605293, −6.33566847138520445969195965773, −5.96302436807233823102114749865, −5.54252969783697792877525410272, −5.15646581384710945016268913876, −4.91117327031861262842107322385, −4.62095972215294935218786942253, −4.39940011913723171627069344898, −3.56306158867635284275151784848, −3.40966950760144177033115637935, −2.91500423146157153529534249782, −2.38166805016640015461763896558, −2.12161146926187361505361991023, −1.62804921666918202782129617088, −1.32815686933421968541508085129, −0.45692153506652390982818887561,
0.45692153506652390982818887561, 1.32815686933421968541508085129, 1.62804921666918202782129617088, 2.12161146926187361505361991023, 2.38166805016640015461763896558, 2.91500423146157153529534249782, 3.40966950760144177033115637935, 3.56306158867635284275151784848, 4.39940011913723171627069344898, 4.62095972215294935218786942253, 4.91117327031861262842107322385, 5.15646581384710945016268913876, 5.54252969783697792877525410272, 5.96302436807233823102114749865, 6.33566847138520445969195965773, 6.75093309700779139201493605293, 6.81927287636335356788331689782, 7.30636815171210682285208297141, 7.54481535332188036567132817036, 8.017216452473982332983097595015
