| L(s) = 1 | + 3·3-s + 2·5-s + 5·7-s + 6·9-s + 6·11-s − 13-s + 6·15-s − 3·17-s + 5·19-s + 15·21-s − 2·23-s − 7·25-s + 9·27-s − 9·29-s + 4·31-s + 18·33-s + 10·35-s − 5·37-s − 3·39-s − 7·41-s + 3·43-s + 12·45-s + 8·47-s + 18·49-s − 9·51-s − 9·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 1.88·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 1.54·15-s − 0.727·17-s + 1.14·19-s + 3.27·21-s − 0.417·23-s − 7/5·25-s + 1.73·27-s − 1.67·29-s + 0.718·31-s + 3.13·33-s + 1.69·35-s − 0.821·37-s − 0.480·39-s − 1.09·41-s + 0.457·43-s + 1.78·45-s + 1.16·47-s + 18/7·49-s − 1.26·51-s − 1.23·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.085342817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.085342817\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.779234120622260103825673264826, −9.718064825195812181491392346196, −9.217845918084275416428514865098, −9.040151919805813430038464962107, −8.572924509691368573195944604944, −8.118943662606817609951073246647, −7.64715858611448764716170142053, −7.58504368879306099563759295735, −6.84199170077041713333442427253, −6.56404412218236368100298542352, −5.74309446452027759966945853436, −5.49046248812668576999836412958, −4.79243623790019665119544464351, −4.35161592462145940197429539428, −3.76790532616847037004619472395, −3.63520668692261658897688127217, −2.60927833244559101138015319216, −2.13329657963466697794068030030, −1.59898483877940763721174448712, −1.39928190328830203230809695440,
1.39928190328830203230809695440, 1.59898483877940763721174448712, 2.13329657963466697794068030030, 2.60927833244559101138015319216, 3.63520668692261658897688127217, 3.76790532616847037004619472395, 4.35161592462145940197429539428, 4.79243623790019665119544464351, 5.49046248812668576999836412958, 5.74309446452027759966945853436, 6.56404412218236368100298542352, 6.84199170077041713333442427253, 7.58504368879306099563759295735, 7.64715858611448764716170142053, 8.118943662606817609951073246647, 8.572924509691368573195944604944, 9.040151919805813430038464962107, 9.217845918084275416428514865098, 9.718064825195812181491392346196, 9.779234120622260103825673264826
