L(s) = 1 | + 2.80·2-s + 3-s + 5.84·4-s + 5-s + 2.80·6-s − 0.809·7-s + 10.7·8-s + 9-s + 2.80·10-s − 4.85·11-s + 5.84·12-s − 13-s − 2.26·14-s + 15-s + 18.5·16-s + 4.54·17-s + 2.80·18-s + 1.06·19-s + 5.84·20-s − 0.809·21-s − 13.6·22-s + 2.82·23-s + 10.7·24-s + 25-s − 2.80·26-s + 27-s − 4.73·28-s + ⋯ |
L(s) = 1 | + 1.98·2-s + 0.577·3-s + 2.92·4-s + 0.447·5-s + 1.14·6-s − 0.305·7-s + 3.81·8-s + 0.333·9-s + 0.885·10-s − 1.46·11-s + 1.68·12-s − 0.277·13-s − 0.605·14-s + 0.258·15-s + 4.62·16-s + 1.10·17-s + 0.660·18-s + 0.245·19-s + 1.30·20-s − 0.176·21-s − 2.90·22-s + 0.589·23-s + 2.20·24-s + 0.200·25-s − 0.549·26-s + 0.192·27-s − 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.35081942\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.35081942\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 7 | \( 1 + 0.809T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 0.457T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 - 1.18T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 9.32T + 61T^{2} \) |
| 67 | \( 1 - 0.998T + 67T^{2} \) |
| 71 | \( 1 + 7.77T + 71T^{2} \) |
| 73 | \( 1 - 4.35T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.78453043675804872049739402324, −7.13222234156529211229441747697, −6.55533332757746545283090973369, −5.45148001485529465846867384738, −5.38867269465107115591263073968, −4.48566367072489243268772766211, −3.56569710884520922543341152458, −2.90638279367113013304406583713, −2.44600443666403100075215423993, −1.38239890153223368585123689547,
1.38239890153223368585123689547, 2.44600443666403100075215423993, 2.90638279367113013304406583713, 3.56569710884520922543341152458, 4.48566367072489243268772766211, 5.38867269465107115591263073968, 5.45148001485529465846867384738, 6.55533332757746545283090973369, 7.13222234156529211229441747697, 7.78453043675804872049739402324