| L(s) = 1 | − 2-s + 2.23·3-s + 4-s − 1.61·5-s − 2.23·6-s + 7-s − 8-s + 2.00·9-s + 1.61·10-s + 2.23·12-s − 4.23·13-s − 14-s − 3.61·15-s + 16-s − 4.23·17-s − 2.00·18-s − 5·19-s − 1.61·20-s + 2.23·21-s + 4.23·23-s − 2.23·24-s − 2.38·25-s + 4.23·26-s − 2.23·27-s + 28-s − 7.47·29-s + 3.61·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.723·5-s − 0.912·6-s + 0.377·7-s − 0.353·8-s + 0.666·9-s + 0.511·10-s + 0.645·12-s − 1.17·13-s − 0.267·14-s − 0.934·15-s + 0.250·16-s − 1.02·17-s − 0.471·18-s − 1.14·19-s − 0.361·20-s + 0.487·21-s + 0.883·23-s − 0.456·24-s − 0.476·25-s + 0.830·26-s − 0.430·27-s + 0.188·28-s − 1.38·29-s + 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.846518020031309699125358122500, −8.332846284800719841138168834491, −7.36505017920155662320087261166, −7.21758282055367036035506599126, −5.78407136152430713178975154679, −4.52107665611775297081404816142, −3.73538249763026529558468602620, −2.60354351653892219918721328524, −1.93298015322277437056280839533, 0,
1.93298015322277437056280839533, 2.60354351653892219918721328524, 3.73538249763026529558468602620, 4.52107665611775297081404816142, 5.78407136152430713178975154679, 7.21758282055367036035506599126, 7.36505017920155662320087261166, 8.332846284800719841138168834491, 8.846518020031309699125358122500
