| L(s) = 1 | + 12·13-s − 8·25-s + 24·37-s + 14·49-s − 24·61-s − 32·73-s − 16·97-s − 12·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.32·13-s − 8/5·25-s + 3.94·37-s + 2·49-s − 3.07·61-s − 3.74·73-s − 1.62·97-s − 1.14·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.016084463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.016084463\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.311605083776251219527285044048, −8.921197137996507792736773708262, −8.353287292785018680158339013856, −8.072034234144190227522712697900, −7.74560525245172010228417041592, −7.43086219006204179904053682752, −6.78339017991032021062723201125, −6.25286859137506142840547469556, −6.09586485774681856113089313094, −5.71011642332748268342188398922, −5.63773615633754931085784872722, −4.58674020228735418102628719951, −4.18336573623561732401111854282, −4.08189914076739796400563549059, −3.56638747554762841304614030293, −2.85783639298055290736943660227, −2.66986177275466672788900327151, −1.54537820146411375290948499326, −1.44122578841076782778970346731, −0.64430929532382185917306103077,
0.64430929532382185917306103077, 1.44122578841076782778970346731, 1.54537820146411375290948499326, 2.66986177275466672788900327151, 2.85783639298055290736943660227, 3.56638747554762841304614030293, 4.08189914076739796400563549059, 4.18336573623561732401111854282, 4.58674020228735418102628719951, 5.63773615633754931085784872722, 5.71011642332748268342188398922, 6.09586485774681856113089313094, 6.25286859137506142840547469556, 6.78339017991032021062723201125, 7.43086219006204179904053682752, 7.74560525245172010228417041592, 8.072034234144190227522712697900, 8.353287292785018680158339013856, 8.921197137996507792736773708262, 9.311605083776251219527285044048
