L(s) = 1 | + 0.185·2-s + 3-s − 1.96·4-s − 0.737·5-s + 0.185·6-s − 0.737·8-s + 9-s − 0.137·10-s − 11-s − 1.96·12-s − 1.45·13-s − 0.737·15-s + 3.79·16-s + 0.694·17-s + 0.185·18-s + 2.62·19-s + 1.44·20-s − 0.185·22-s + 4.77·23-s − 0.737·24-s − 4.45·25-s − 0.270·26-s + 27-s + 0.434·29-s − 0.137·30-s + 2.04·31-s + 2.17·32-s + ⋯ |
L(s) = 1 | + 0.131·2-s + 0.577·3-s − 0.982·4-s − 0.329·5-s + 0.0758·6-s − 0.260·8-s + 0.333·9-s − 0.0433·10-s − 0.301·11-s − 0.567·12-s − 0.404·13-s − 0.190·15-s + 0.948·16-s + 0.168·17-s + 0.0438·18-s + 0.602·19-s + 0.323·20-s − 0.0396·22-s + 0.996·23-s − 0.150·24-s − 0.891·25-s − 0.0531·26-s + 0.192·27-s + 0.0806·29-s − 0.0250·30-s + 0.367·31-s + 0.385·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574615283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574615283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.185T + 2T^{2} \) |
| 5 | \( 1 + 0.737T + 5T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 - 0.694T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 0.434T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 - 9.34T + 41T^{2} \) |
| 43 | \( 1 + 0.108T + 43T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 + 0.877T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 0.0977T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 5.71T + 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−9.435063826715740402865395395056, −8.605537607947202133354086278630, −7.87059758277812690437673893679, −7.28433511185877635998451921623, −6.01865732249670172302942409181, −5.11693929078611596232838008983, −4.32900601222738931892621696250, −3.49485213195735654362476046004, −2.52214562578352066865647679159, −0.856013403955148926390061490366,
0.856013403955148926390061490366, 2.52214562578352066865647679159, 3.49485213195735654362476046004, 4.32900601222738931892621696250, 5.11693929078611596232838008983, 6.01865732249670172302942409181, 7.28433511185877635998451921623, 7.87059758277812690437673893679, 8.605537607947202133354086278630, 9.435063826715740402865395395056