| L(s) = 1 | + 2.60·2-s + 4.77·4-s − 2.74·5-s − 2.14·7-s + 7.23·8-s − 7.15·10-s + 0.935·11-s − 3.80·13-s − 5.58·14-s + 9.26·16-s + 0.395·17-s + 3.48·19-s − 13.1·20-s + 2.43·22-s + 23-s + 2.55·25-s − 9.91·26-s − 10.2·28-s + 29-s − 6.61·31-s + 9.67·32-s + 1.02·34-s + 5.89·35-s − 9.78·37-s + 9.08·38-s − 19.8·40-s − 9.64·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.38·4-s − 1.22·5-s − 0.810·7-s + 2.55·8-s − 2.26·10-s + 0.281·11-s − 1.05·13-s − 1.49·14-s + 2.31·16-s + 0.0958·17-s + 0.800·19-s − 2.93·20-s + 0.519·22-s + 0.208·23-s + 0.511·25-s − 1.94·26-s − 1.93·28-s + 0.185·29-s − 1.18·31-s + 1.70·32-s + 0.176·34-s + 0.996·35-s − 1.60·37-s + 1.47·38-s − 3.14·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 0.935T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 - 0.395T + 17T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 + 9.78T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 6.39T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 + 0.642T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 3.20T + 79T^{2} \) |
| 83 | \( 1 + 0.860T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.40846825869724186405171769984, −6.87413633528585424027403969628, −6.28159679890738204778996446260, −5.23099090078532283409903905980, −4.90274235618057916173969968994, −3.91836773692749726605054037600, −3.44753660285780656176266471982, −2.88872727800454842980602848370, −1.73740181225608367248466492218, 0,
1.73740181225608367248466492218, 2.88872727800454842980602848370, 3.44753660285780656176266471982, 3.91836773692749726605054037600, 4.90274235618057916173969968994, 5.23099090078532283409903905980, 6.28159679890738204778996446260, 6.87413633528585424027403969628, 7.40846825869724186405171769984
