| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 9-s − 2·11-s + 14-s + 16-s − 18-s − 2·22-s + 8·23-s + 2·25-s + 28-s − 4·29-s + 32-s − 36-s − 4·37-s + 10·43-s − 2·44-s + 8·46-s + 49-s + 2·50-s + 4·53-s + 56-s − 4·58-s − 63-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1/3·9-s − 0.603·11-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.426·22-s + 1.66·23-s + 2/5·25-s + 0.188·28-s − 0.742·29-s + 0.176·32-s − 1/6·36-s − 0.657·37-s + 1.52·43-s − 0.301·44-s + 1.17·46-s + 1/7·49-s + 0.282·50-s + 0.549·53-s + 0.133·56-s − 0.525·58-s − 0.125·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.187998900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.187998900\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
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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
−8.245858188636532214425302952902, −7.59191336140147705701907189374, −7.36946961833629769985140310897, −7.01255249572986766938047727350, −6.32538739078004097013212769631, −5.99099274768105546060922165486, −5.32459913587493009453196778277, −5.16093512383294595206664755704, −4.63439090825865879709001074675, −4.05731046548562162883819863707, −3.48090943650656560617513989175, −2.92306066446628594165927470923, −2.44991724932004317342586916735, −1.72258213665233259919558956932, −0.78113042585714534495479530171,
0.78113042585714534495479530171, 1.72258213665233259919558956932, 2.44991724932004317342586916735, 2.92306066446628594165927470923, 3.48090943650656560617513989175, 4.05731046548562162883819863707, 4.63439090825865879709001074675, 5.16093512383294595206664755704, 5.32459913587493009453196778277, 5.99099274768105546060922165486, 6.32538739078004097013212769631, 7.01255249572986766938047727350, 7.36946961833629769985140310897, 7.59191336140147705701907189374, 8.245858188636532214425302952902
