| 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 − 8·29-s − 32-s + 36-s + 12·37-s + 2·43-s + 2·44-s − 8·46-s + 49-s + 2·50-s + 4·53-s + 56-s + 8·58-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 − 1.48·29-s − 0.176·32-s + 1/6·36-s + 1.97·37-s + 0.304·43-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 + 1.05·58-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
| \(L(1)\) |
\(\approx\) |
\(1.434526316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434526316\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 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$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.263413389218204735046809748611, −7.77581627482105251330440321366, −7.36090497095372997849663828717, −7.04383274262450592305217368595, −6.57630274507430510175427601787, −6.05242183910978406458714315405, −5.72021454179047845573286447900, −5.06748465502921289624588695665, −4.54549395475276678198165379958, −3.88998866166277692783304988293, −3.51143383008453470965805560648, −2.78334114942570401939101098187, −2.24146060442707014941072038849, −1.42406492024740682558352108186, −0.69437058197589203950632984813,
0.69437058197589203950632984813, 1.42406492024740682558352108186, 2.24146060442707014941072038849, 2.78334114942570401939101098187, 3.51143383008453470965805560648, 3.88998866166277692783304988293, 4.54549395475276678198165379958, 5.06748465502921289624588695665, 5.72021454179047845573286447900, 6.05242183910978406458714315405, 6.57630274507430510175427601787, 7.04383274262450592305217368595, 7.36090497095372997849663828717, 7.77581627482105251330440321366, 8.263413389218204735046809748611
