L(s) = 1 | − 2.15·2-s − 0.292·3-s + 2.64·4-s + 5-s + 0.629·6-s − 5.08·7-s − 1.39·8-s − 2.91·9-s − 2.15·10-s − 2.47·11-s − 0.772·12-s − 1.31·13-s + 10.9·14-s − 0.292·15-s − 2.29·16-s + 17-s + 6.28·18-s + 4.05·19-s + 2.64·20-s + 1.48·21-s + 5.32·22-s + 3.69·23-s + 0.406·24-s + 25-s + 2.83·26-s + 1.72·27-s − 13.4·28-s + ⋯ |
L(s) = 1 | − 1.52·2-s − 0.168·3-s + 1.32·4-s + 0.447·5-s + 0.256·6-s − 1.92·7-s − 0.492·8-s − 0.971·9-s − 0.681·10-s − 0.744·11-s − 0.223·12-s − 0.364·13-s + 2.92·14-s − 0.0753·15-s − 0.572·16-s + 0.242·17-s + 1.48·18-s + 0.930·19-s + 0.591·20-s + 0.323·21-s + 1.13·22-s + 0.771·23-s + 0.0829·24-s + 0.200·25-s + 0.555·26-s + 0.332·27-s − 2.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.292T + 3T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 + 2.91T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 - 5.91T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 2.64T + 53T^{2} \) |
| 59 | \( 1 - 0.477T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 6.06T + 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.79997642570414523700102140499, −7.04506665976398484901420506335, −6.63845375591529186434269753240, −5.67394666497822834353905077373, −5.24369198572904076606364827607, −3.70159933222540792794318519436, −2.88703337842518036422059105188, −2.30351721416963306780155447293, −0.847764109001253295085835062047, 0,
0.847764109001253295085835062047, 2.30351721416963306780155447293, 2.88703337842518036422059105188, 3.70159933222540792794318519436, 5.24369198572904076606364827607, 5.67394666497822834353905077373, 6.63845375591529186434269753240, 7.04506665976398484901420506335, 7.79997642570414523700102140499