| L(s) = 1 | + 2·3-s + 3·9-s + 6·13-s − 4·17-s + 6·25-s + 4·27-s − 12·29-s + 12·39-s + 16·43-s − 49-s − 8·51-s − 20·53-s + 4·61-s + 12·75-s + 5·81-s − 24·87-s − 4·101-s − 8·103-s − 8·107-s − 4·113-s + 18·117-s + 18·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.66·13-s − 0.970·17-s + 6/5·25-s + 0.769·27-s − 2.22·29-s + 1.92·39-s + 2.43·43-s − 1/7·49-s − 1.12·51-s − 2.74·53-s + 0.512·61-s + 1.38·75-s + 5/9·81-s − 2.57·87-s − 0.398·101-s − 0.788·103-s − 0.773·107-s − 0.376·113-s + 1.66·117-s + 1.63·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.680562999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.680562999\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.509713573470807653434256579484, −8.239562765179171750397323878534, −7.934259049258317269465534437607, −7.51592220573300675921252993119, −7.12385757039983035624970315250, −6.87137170534161698467125585692, −6.26098351492999928842690339566, −6.20034999096613365337008883500, −5.52049967265216946550847105585, −5.36373200045031844200833228933, −4.59040771198122615419210798264, −4.27905115313627919340122263135, −4.05479813470241852010774543379, −3.51379356435242115963120565580, −3.06611376401934775807416712826, −2.90381404963911294794527739692, −1.96740174532836824700239211849, −1.95646688924764499197606056272, −1.24818228226118868466627152030, −0.57491819717777231768617365656,
0.57491819717777231768617365656, 1.24818228226118868466627152030, 1.95646688924764499197606056272, 1.96740174532836824700239211849, 2.90381404963911294794527739692, 3.06611376401934775807416712826, 3.51379356435242115963120565580, 4.05479813470241852010774543379, 4.27905115313627919340122263135, 4.59040771198122615419210798264, 5.36373200045031844200833228933, 5.52049967265216946550847105585, 6.20034999096613365337008883500, 6.26098351492999928842690339566, 6.87137170534161698467125585692, 7.12385757039983035624970315250, 7.51592220573300675921252993119, 7.934259049258317269465534437607, 8.239562765179171750397323878534, 8.509713573470807653434256579484
