L(s) = 1 | + 2.11·2-s + 3-s + 2.48·4-s − 5-s + 2.11·6-s + 3.62·7-s + 1.03·8-s + 9-s − 2.11·10-s − 1.61·11-s + 2.48·12-s + 5.62·13-s + 7.68·14-s − 15-s − 2.78·16-s − 4.26·17-s + 2.11·18-s + 5.99·19-s − 2.48·20-s + 3.62·21-s − 3.43·22-s + 2.33·23-s + 1.03·24-s + 25-s + 11.9·26-s + 27-s + 9.03·28-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.447·5-s + 0.864·6-s + 1.37·7-s + 0.366·8-s + 0.333·9-s − 0.669·10-s − 0.488·11-s + 0.718·12-s + 1.56·13-s + 2.05·14-s − 0.258·15-s − 0.695·16-s − 1.03·17-s + 0.499·18-s + 1.37·19-s − 0.556·20-s + 0.791·21-s − 0.731·22-s + 0.485·23-s + 0.211·24-s + 0.200·25-s + 2.33·26-s + 0.192·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.694692702\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.694692702\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 5.62T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 0.885T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 + 5.13T + 61T^{2} \) |
| 67 | \( 1 + 0.936T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 - 2.19T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 6.09T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−8.055052939286555536844115571881, −7.25580392872975207059517193155, −6.59422485338728152460492650511, −5.65373715891640700957934236383, −5.04407226415113111307978338347, −4.43193205371262771426469783801, −3.74957194049797513608120601461, −3.06315090383731067215722634429, −2.17816536460168432128746259607, −1.14967188520436726621771209824,
1.14967188520436726621771209824, 2.17816536460168432128746259607, 3.06315090383731067215722634429, 3.74957194049797513608120601461, 4.43193205371262771426469783801, 5.04407226415113111307978338347, 5.65373715891640700957934236383, 6.59422485338728152460492650511, 7.25580392872975207059517193155, 8.055052939286555536844115571881