L(s) = 1 | − 2.59·2-s − 3-s + 4.71·4-s + 2.59·6-s − 3.41·7-s − 7.04·8-s + 9-s − 4.71·12-s − 4.58·13-s + 8.86·14-s + 8.82·16-s − 1.97·17-s − 2.59·18-s − 1.41·19-s + 3.41·21-s + 5.03·23-s + 7.04·24-s + 11.8·26-s − 27-s − 16.1·28-s − 4.34·29-s − 0.273·31-s − 8.78·32-s + 5.12·34-s + 4.71·36-s − 0.245·37-s + 3.67·38-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.35·4-s + 1.05·6-s − 1.29·7-s − 2.49·8-s + 0.333·9-s − 1.36·12-s − 1.27·13-s + 2.36·14-s + 2.20·16-s − 0.479·17-s − 0.610·18-s − 0.325·19-s + 0.745·21-s + 1.04·23-s + 1.43·24-s + 2.32·26-s − 0.192·27-s − 3.04·28-s − 0.805·29-s − 0.0490·31-s − 1.55·32-s + 0.879·34-s + 0.786·36-s − 0.0403·37-s + 0.596·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 5.03T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 0.273T + 31T^{2} \) |
| 37 | \( 1 + 0.245T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 0.972T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 9.09T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.769T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.39890965620057627488401136720, −6.83822705517118685160640779176, −6.45136416343493257445851908228, −5.63266848355493570660327740142, −4.73429861062293698615000630548, −3.55282680069293123161941128245, −2.68186213835170015709411281534, −1.99241600330231198848337102202, −0.76189806156121308869609513728, 0,
0.76189806156121308869609513728, 1.99241600330231198848337102202, 2.68186213835170015709411281534, 3.55282680069293123161941128245, 4.73429861062293698615000630548, 5.63266848355493570660327740142, 6.45136416343493257445851908228, 6.83822705517118685160640779176, 7.39890965620057627488401136720