| L(s) = 1 | − 0.847·2-s + 0.0176·3-s − 1.28·4-s + 3.43·5-s − 0.0149·6-s + 0.269·7-s + 2.78·8-s − 2.99·9-s − 2.91·10-s + 3.29·11-s − 0.0226·12-s − 2.01·13-s − 0.227·14-s + 0.0606·15-s + 0.207·16-s − 6.44·17-s + 2.54·18-s − 6.01·19-s − 4.40·20-s + 0.00474·21-s − 2.79·22-s − 2.22·23-s + 0.0490·24-s + 6.81·25-s + 1.70·26-s − 0.105·27-s − 0.344·28-s + ⋯ |
| L(s) = 1 | − 0.599·2-s + 0.0101·3-s − 0.640·4-s + 1.53·5-s − 0.00610·6-s + 0.101·7-s + 0.983·8-s − 0.999·9-s − 0.920·10-s + 0.993·11-s − 0.00652·12-s − 0.559·13-s − 0.0609·14-s + 0.0156·15-s + 0.0517·16-s − 1.56·17-s + 0.599·18-s − 1.37·19-s − 0.985·20-s + 0.00103·21-s − 0.595·22-s − 0.463·23-s + 0.0100·24-s + 1.36·25-s + 0.335·26-s − 0.0203·27-s − 0.0651·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.847T + 2T^{2} \) |
| 3 | \( 1 - 0.0176T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 0.269T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 6.03T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.64T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−8.935483218867842057042169525168, −8.539977862533866968315283642123, −7.27869027540780563672628206088, −6.38010658431149317770640792378, −5.73607012877964370828755985992, −4.81727000182828004630395620085, −3.93945733636418274778373739806, −2.40226520107546377552394770771, −1.69119979488966231970219904971, 0,
1.69119979488966231970219904971, 2.40226520107546377552394770771, 3.93945733636418274778373739806, 4.81727000182828004630395620085, 5.73607012877964370828755985992, 6.38010658431149317770640792378, 7.27869027540780563672628206088, 8.539977862533866968315283642123, 8.935483218867842057042169525168
