L(s) = 1 | + 3-s − 2·5-s − 2·7-s − 9-s − 11-s − 13-s − 2·15-s + 11·17-s − 6·19-s − 2·21-s + 2·23-s + 3·25-s − 5·29-s + 4·31-s − 33-s + 4·35-s − 39-s + 6·41-s − 6·43-s + 2·45-s − 9·47-s + 3·49-s + 11·51-s + 18·53-s + 2·55-s − 6·57-s − 8·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s − 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s + 2.66·17-s − 1.37·19-s − 0.436·21-s + 0.417·23-s + 3/5·25-s − 0.928·29-s + 0.718·31-s − 0.174·33-s + 0.676·35-s − 0.160·39-s + 0.937·41-s − 0.914·43-s + 0.298·45-s − 1.31·47-s + 3/7·49-s + 1.54·51-s + 2.47·53-s + 0.269·55-s − 0.794·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.867397775\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867397775\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 150 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.097234647893136867138292200782, −8.888440225585783614233151823523, −8.273949359172296328764915601089, −8.148981601051169129242443610138, −7.76186171553721632148810553126, −7.43041258217996497388306100669, −6.93661388526730255723923729506, −6.65015537040905709609359887573, −5.95270970983218109135960554127, −5.85453034255477111858631590947, −5.15379440630360228346259705924, −4.92335186230312731711724586326, −4.29059403190164365663500040068, −3.72360740884641937350703114632, −3.32901813046669970016404063607, −3.29793327922374620726240486102, −2.44132100443288269508840438101, −2.17126911937498609265083742310, −1.09719484108411733098410469426, −0.52246907753058561017050957950,
0.52246907753058561017050957950, 1.09719484108411733098410469426, 2.17126911937498609265083742310, 2.44132100443288269508840438101, 3.29793327922374620726240486102, 3.32901813046669970016404063607, 3.72360740884641937350703114632, 4.29059403190164365663500040068, 4.92335186230312731711724586326, 5.15379440630360228346259705924, 5.85453034255477111858631590947, 5.95270970983218109135960554127, 6.65015537040905709609359887573, 6.93661388526730255723923729506, 7.43041258217996497388306100669, 7.76186171553721632148810553126, 8.148981601051169129242443610138, 8.273949359172296328764915601089, 8.888440225585783614233151823523, 9.097234647893136867138292200782