L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 17-s − 18-s − 2·19-s − 25-s − 32-s + 34-s + 36-s + 2·38-s + 2·43-s − 49-s + 50-s + 2·59-s + 64-s − 2·67-s − 68-s − 72-s − 2·76-s + 81-s + 2·83-s − 2·86-s − 2·89-s + 98-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 17-s − 18-s − 2·19-s − 25-s − 32-s + 34-s + 36-s + 2·38-s + 2·43-s − 49-s + 50-s + 2·59-s + 64-s − 2·67-s − 68-s − 72-s − 2·76-s + 81-s + 2·83-s − 2·86-s − 2·89-s + 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4342953204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4342953204\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.24083381227856928189287299030, −12.37060503711264230626462419909, −11.12990696280955116118199910364, −10.34652595840034658900516186202, −9.309527469753088104438031600551, −8.296626719201967746883695806752, −7.14365748685889483187532804768, −6.16344603975208278087909120617, −4.20651178016227437261911449698, −2.11561848960962061958698046486,
2.11561848960962061958698046486, 4.20651178016227437261911449698, 6.16344603975208278087909120617, 7.14365748685889483187532804768, 8.296626719201967746883695806752, 9.309527469753088104438031600551, 10.34652595840034658900516186202, 11.12990696280955116118199910364, 12.37060503711264230626462419909, 13.24083381227856928189287299030