| L(s) = 1 | − 3-s + 9-s + 19-s − 25-s − 27-s + 2·41-s + 2·43-s + 49-s − 57-s + 2·59-s − 2·73-s + 75-s + 81-s − 2·89-s + 2·107-s + 2·113-s + ⋯ |
| L(s) = 1 | − 3-s + 9-s + 19-s − 25-s − 27-s + 2·41-s + 2·43-s + 49-s − 57-s + 2·59-s − 2·73-s + 75-s + 81-s − 2·89-s + 2·107-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8472890666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8472890666\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.633089832413747717694042659446, −8.756644988815678996458518121891, −7.55316134767504168863775602216, −7.21551138122331369137622763808, −5.97343330464111443839884741849, −5.65750651021129222845256008155, −4.56059810791229171007035493168, −3.80662040205189233910342903474, −2.42157440110287334733486592433, −1.01825585187034011117862883853,
1.01825585187034011117862883853, 2.42157440110287334733486592433, 3.80662040205189233910342903474, 4.56059810791229171007035493168, 5.65750651021129222845256008155, 5.97343330464111443839884741849, 7.21551138122331369137622763808, 7.55316134767504168863775602216, 8.756644988815678996458518121891, 9.633089832413747717694042659446
