| L(s) = 1 | − 3-s + 4-s − 7-s + 9-s − 12-s + 13-s + 16-s + 21-s + 25-s − 27-s − 28-s + 31-s + 36-s + 37-s − 39-s − 43-s − 48-s + 52-s − 61-s − 63-s + 64-s + 67-s − 73-s − 75-s + 79-s + 81-s + 84-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 7-s + 9-s − 12-s + 13-s + 16-s + 21-s + 25-s − 27-s − 28-s + 31-s + 36-s + 37-s − 39-s − 43-s − 48-s + 52-s − 61-s − 63-s + 64-s + 67-s − 73-s − 75-s + 79-s + 81-s + 84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9312040032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9312040032\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25728976396643285983335711719, −9.522582210852831875785313574965, −8.328420537264900266056321133544, −7.30030504110544656493729661373, −6.42065871048687356454484497929, −6.21325436035647197432696166228, −5.08975614828693295823032851891, −3.83133567278554226786437890612, −2.79786434434247274373228835084, −1.26254058663402173676258643870,
1.26254058663402173676258643870, 2.79786434434247274373228835084, 3.83133567278554226786437890612, 5.08975614828693295823032851891, 6.21325436035647197432696166228, 6.42065871048687356454484497929, 7.30030504110544656493729661373, 8.328420537264900266056321133544, 9.522582210852831875785313574965, 10.25728976396643285983335711719
