| L(s) = 1 | − 8·7-s + 2·9-s + 88·23-s − 25·25-s − 124·41-s + 8·47-s − 50·49-s − 16·63-s − 77·81-s + 284·89-s + 88·103-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 704·161-s + 163-s + 167-s − 338·169-s + 173-s + 200·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 8/7·7-s + 2/9·9-s + 3.82·23-s − 25-s − 3.02·41-s + 8/47·47-s − 1.02·49-s − 0.253·63-s − 0.950·81-s + 3.19·89-s + 0.854·103-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 4.37·161-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 8/7·175-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.485150799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485150799\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1198 T^{2} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2078 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 4078 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4478 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8002 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−9.709522022090276329098563625636, −9.123698712108264881233629007433, −9.068973803963312490243754942738, −8.564704776049152732813003457580, −8.129430215047166191015492150163, −7.38110465145316162593747304437, −7.32895438964099767780841807848, −6.75858471785717614873811513257, −6.33636104148734428505017865299, −6.24863816301040650669559869645, −5.18702157200992367309547424656, −5.11922337529695829585761535868, −4.80844599379100670912543286831, −3.80199478947581832955197409032, −3.61863463185476116642467749967, −2.96178156975455524157690426332, −2.76581517558307102902251633139, −1.77674798588697002448546208097, −1.21527649783633792059533219864, −0.37852056963888892286509842834,
0.37852056963888892286509842834, 1.21527649783633792059533219864, 1.77674798588697002448546208097, 2.76581517558307102902251633139, 2.96178156975455524157690426332, 3.61863463185476116642467749967, 3.80199478947581832955197409032, 4.80844599379100670912543286831, 5.11922337529695829585761535868, 5.18702157200992367309547424656, 6.24863816301040650669559869645, 6.33636104148734428505017865299, 6.75858471785717614873811513257, 7.32895438964099767780841807848, 7.38110465145316162593747304437, 8.129430215047166191015492150163, 8.564704776049152732813003457580, 9.068973803963312490243754942738, 9.123698712108264881233629007433, 9.709522022090276329098563625636
