| L(s) = 1 | + 4·7-s − 3·13-s + 5·25-s − 18·31-s + 2·37-s + 8·43-s + 9·49-s + 15·61-s − 11·67-s + 13·79-s − 12·91-s − 33·97-s − 33·103-s − 4·109-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7·169-s + 173-s + 20·175-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.832·13-s + 25-s − 3.23·31-s + 0.328·37-s + 1.21·43-s + 9/7·49-s + 1.92·61-s − 1.34·67-s + 1.46·79-s − 1.25·91-s − 3.35·97-s − 3.25·103-s − 0.383·109-s − 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 − 0.538·169-s + 0.0760·173-s + 1.51·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432512266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432512266\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.271998846550103101975878626562, −8.876609393762842801708941162212, −8.290516287761940316948285894917, −8.058591577841296844896059246569, −7.83403709587226284804162897743, −7.12119035389214165640408718479, −6.99701297045068367672881413887, −6.82824316388377831053822748708, −5.78818784945066624681164625895, −5.50574558097363266176893818596, −5.48134602866110470966231110422, −4.79553712772513149935643037734, −4.39482547467703296118627195461, −4.09962165014308593599537602616, −3.49081813135816885939045537244, −2.92199407609548793006755924537, −2.33958643739715296745029178274, −1.88349680519783998838227727060, −1.40258669156789020295432180216, −0.54459385875799486262768981193,
0.54459385875799486262768981193, 1.40258669156789020295432180216, 1.88349680519783998838227727060, 2.33958643739715296745029178274, 2.92199407609548793006755924537, 3.49081813135816885939045537244, 4.09962165014308593599537602616, 4.39482547467703296118627195461, 4.79553712772513149935643037734, 5.48134602866110470966231110422, 5.50574558097363266176893818596, 5.78818784945066624681164625895, 6.82824316388377831053822748708, 6.99701297045068367672881413887, 7.12119035389214165640408718479, 7.83403709587226284804162897743, 8.058591577841296844896059246569, 8.290516287761940316948285894917, 8.876609393762842801708941162212, 9.271998846550103101975878626562
