L(s) = 1 | + 0.423·2-s + 3-s − 1.82·4-s − 1.36·5-s + 0.423·6-s + 0.865·7-s − 1.61·8-s + 9-s − 0.577·10-s − 11-s − 1.82·12-s + 2.63·13-s + 0.366·14-s − 1.36·15-s + 2.95·16-s − 3.12·17-s + 0.423·18-s − 0.281·19-s + 2.48·20-s + 0.865·21-s − 0.423·22-s + 1.73·23-s − 1.61·24-s − 3.13·25-s + 1.11·26-s + 27-s − 1.57·28-s + ⋯ |
L(s) = 1 | + 0.299·2-s + 0.577·3-s − 0.910·4-s − 0.610·5-s + 0.172·6-s + 0.327·7-s − 0.571·8-s + 0.333·9-s − 0.182·10-s − 0.301·11-s − 0.525·12-s + 0.731·13-s + 0.0978·14-s − 0.352·15-s + 0.739·16-s − 0.758·17-s + 0.0997·18-s − 0.0646·19-s + 0.555·20-s + 0.188·21-s − 0.0902·22-s + 0.362·23-s − 0.329·24-s − 0.627·25-s + 0.218·26-s + 0.192·27-s − 0.297·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 0.423T + 2T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 0.865T + 7T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 0.281T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 67 | \( 1 - 4.80T + 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 6.03T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.801182196745599719492747932568, −8.025315504351552006939727536747, −7.47561088498641935933059723427, −6.31060224618463206245081864428, −5.39335777798924193731996540751, −4.49241596427811710014009028096, −3.85723143127192489627914613661, −3.05943028939640378825534977570, −1.65634094120654582061591255767, 0,
1.65634094120654582061591255767, 3.05943028939640378825534977570, 3.85723143127192489627914613661, 4.49241596427811710014009028096, 5.39335777798924193731996540751, 6.31060224618463206245081864428, 7.47561088498641935933059723427, 8.025315504351552006939727536747, 8.801182196745599719492747932568