| L(s) = 1 | + 4·17-s − 50·25-s − 92·41-s + 98·49-s − 284·73-s − 292·89-s + 188·97-s − 196·113-s − 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4/17·17-s − 2·25-s − 2.24·41-s + 2·49-s − 3.89·73-s − 3.28·89-s + 1.93·97-s − 1.73·113-s − 0.380·121-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 + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1999632999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1999632999\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )( 1 + 62 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 158 T + p^{2} T^{2} )( 1 + 158 T + p^{2} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−9.235384836471444207042729171731, −8.454812691377401612463180879601, −8.448139312718871238150161127569, −7.78625039234257766624864220524, −7.52720701895520787846921348283, −7.08125241648226899241500919212, −6.83616088820026541030727861164, −6.05503360488972100009877396828, −6.05175578867713213553138427863, −5.38520497966722607213080345193, −5.27411531435400442767236022139, −4.37285108708539719384653675937, −4.35655484238531751557438516364, −3.62108881631408503189877016112, −3.41815580981282707536706100955, −2.63214251248987863495303488788, −2.33839353386164092967432426258, −1.53426115256905361727109725381, −1.27201985630527599185413888042, −0.10814679148167641565044134418,
0.10814679148167641565044134418, 1.27201985630527599185413888042, 1.53426115256905361727109725381, 2.33839353386164092967432426258, 2.63214251248987863495303488788, 3.41815580981282707536706100955, 3.62108881631408503189877016112, 4.35655484238531751557438516364, 4.37285108708539719384653675937, 5.27411531435400442767236022139, 5.38520497966722607213080345193, 6.05175578867713213553138427863, 6.05503360488972100009877396828, 6.83616088820026541030727861164, 7.08125241648226899241500919212, 7.52720701895520787846921348283, 7.78625039234257766624864220524, 8.448139312718871238150161127569, 8.454812691377401612463180879601, 9.235384836471444207042729171731
