| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 2·12-s + 13-s + 16-s + 23-s − 25-s − 4·27-s − 2·29-s + 3·36-s − 2·39-s − 2·48-s − 49-s + 52-s + 64-s − 2·69-s + 2·75-s + 5·81-s + 4·87-s + 92-s − 100-s − 2·101-s − 4·108-s − 2·116-s + 3·117-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 2·12-s + 13-s + 16-s + 23-s − 25-s − 4·27-s − 2·29-s + 3·36-s − 2·39-s − 2·48-s − 49-s + 52-s + 64-s − 2·69-s + 2·75-s + 5·81-s + 4·87-s + 92-s − 100-s − 2·101-s − 4·108-s − 2·116-s + 3·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5721878003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5721878003\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 + T )^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.63694941026254215634808843385, −11.19683172004358634714937892688, −10.56771224122569879755281874751, −9.531633927663162276944355375257, −7.70799845773071589574109374536, −6.82686335654376355513749251584, −6.03209179338727940790172675075, −5.30518353834462847992448071362, −3.84481664584623249338008754536, −1.55388557677257811759753769981,
1.55388557677257811759753769981, 3.84481664584623249338008754536, 5.30518353834462847992448071362, 6.03209179338727940790172675075, 6.82686335654376355513749251584, 7.70799845773071589574109374536, 9.531633927663162276944355375257, 10.56771224122569879755281874751, 11.19683172004358634714937892688, 11.63694941026254215634808843385
