| L(s) = 1 | − 3-s + 4-s − 2·5-s − 2·7-s − 4·9-s − 12-s + 7·13-s + 2·15-s − 3·16-s + 9·17-s − 2·19-s − 2·20-s + 2·21-s + 3·25-s + 6·27-s − 2·28-s − 5·29-s + 2·31-s + 4·35-s − 4·36-s − 10·37-s − 7·39-s − 4·41-s − 6·43-s + 8·45-s − 9·47-s + 3·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 4/3·9-s − 0.288·12-s + 1.94·13-s + 0.516·15-s − 3/4·16-s + 2.18·17-s − 0.458·19-s − 0.447·20-s + 0.436·21-s + 3/5·25-s + 1.15·27-s − 0.377·28-s − 0.928·29-s + 0.359·31-s + 0.676·35-s − 2/3·36-s − 1.64·37-s − 1.12·39-s − 0.624·41-s − 0.914·43-s + 1.19·45-s − 1.31·47-s + 0.433·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15 T + 201 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + 65 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 189 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.108849274358393176865383768812, −7.83381181161921086560740097402, −7.72417322311633177595330347068, −6.91222044819744054140569307052, −6.68154807611888045784429879910, −6.29283430302397089706110014698, −6.21384485473736703666626956045, −5.66530436588570559453398090770, −5.21606975307320167704675252413, −5.10681311983347485691623735886, −4.40220651985063147806825784686, −3.88712960122209134110969594404, −3.35557177450042100729634146255, −3.23856495992822729425301149416, −3.09121394664575965841304717626, −2.19808259691164166231137285095, −1.53020755883409821262301194313, −1.13806523522405957672232335676, 0, 0,
1.13806523522405957672232335676, 1.53020755883409821262301194313, 2.19808259691164166231137285095, 3.09121394664575965841304717626, 3.23856495992822729425301149416, 3.35557177450042100729634146255, 3.88712960122209134110969594404, 4.40220651985063147806825784686, 5.10681311983347485691623735886, 5.21606975307320167704675252413, 5.66530436588570559453398090770, 6.21384485473736703666626956045, 6.29283430302397089706110014698, 6.68154807611888045784429879910, 6.91222044819744054140569307052, 7.72417322311633177595330347068, 7.83381181161921086560740097402, 8.108849274358393176865383768812
