L(s) = 1 | + 2-s − 1.55·3-s + 4-s + 2.63·5-s − 1.55·6-s − 0.882·7-s + 8-s − 0.589·9-s + 2.63·10-s + 5.91·11-s − 1.55·12-s − 0.542·13-s − 0.882·14-s − 4.08·15-s + 16-s − 4.94·17-s − 0.589·18-s − 6.50·19-s + 2.63·20-s + 1.36·21-s + 5.91·22-s − 5.08·23-s − 1.55·24-s + 1.92·25-s − 0.542·26-s + 5.57·27-s − 0.882·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.896·3-s + 0.5·4-s + 1.17·5-s − 0.633·6-s − 0.333·7-s + 0.353·8-s − 0.196·9-s + 0.832·10-s + 1.78·11-s − 0.448·12-s − 0.150·13-s − 0.235·14-s − 1.05·15-s + 0.250·16-s − 1.19·17-s − 0.138·18-s − 1.49·19-s + 0.588·20-s + 0.298·21-s + 1.26·22-s − 1.05·23-s − 0.316·24-s + 0.385·25-s − 0.106·26-s + 1.07·27-s − 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
good | 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 + 0.882T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 + 0.542T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 + 6.50T + 19T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 - 9.49T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 - 6.91T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 + 7.41T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−6.77695294145501444597138545010, −6.58866761549132113015049685968, −6.22699702366823447490600522105, −5.51046688845319017743666473418, −4.68401955874735400769791904367, −4.14746767346310710922954670030, −3.15042499565589102592572352247, −2.12769516465602222163709773118, −1.50701397725177967149228419846, 0,
1.50701397725177967149228419846, 2.12769516465602222163709773118, 3.15042499565589102592572352247, 4.14746767346310710922954670030, 4.68401955874735400769791904367, 5.51046688845319017743666473418, 6.22699702366823447490600522105, 6.58866761549132113015049685968, 6.77695294145501444597138545010