| L(s) = 1 | − 1.05·2-s − 0.895·4-s − 3.10·5-s + 3.07·7-s + 3.04·8-s + 3.26·10-s + 3.20·11-s − 0.0429·13-s − 3.23·14-s − 1.40·16-s + 4.01·17-s − 7.31·19-s + 2.78·20-s − 3.36·22-s − 23-s + 4.64·25-s + 0.0451·26-s − 2.75·28-s − 29-s + 0.714·31-s − 4.60·32-s − 4.21·34-s − 9.54·35-s − 6.91·37-s + 7.69·38-s − 9.45·40-s + 5.11·41-s + ⋯ |
| L(s) = 1 | − 0.743·2-s − 0.447·4-s − 1.38·5-s + 1.16·7-s + 1.07·8-s + 1.03·10-s + 0.966·11-s − 0.0119·13-s − 0.863·14-s − 0.351·16-s + 0.973·17-s − 1.67·19-s + 0.622·20-s − 0.718·22-s − 0.208·23-s + 0.929·25-s + 0.00886·26-s − 0.520·28-s − 0.185·29-s + 0.128·31-s − 0.814·32-s − 0.723·34-s − 1.61·35-s − 1.13·37-s + 1.24·38-s − 1.49·40-s + 0.798·41-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 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 + 0.0429T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 31 | \( 1 - 0.714T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 - 3.65T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 0.736T + 97T^{2} \) |
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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
−7.978564281797096530197890335437, −7.30927763073737539225285904805, −6.58503742233429749451290233053, −5.41693810532010684114592369260, −4.63119879272535626664041260828, −4.09693081322539557488827564018, −3.50967282752171189775356198957, −1.97261959485721122703504249340, −1.11653145583880370447735726505, 0,
1.11653145583880370447735726505, 1.97261959485721122703504249340, 3.50967282752171189775356198957, 4.09693081322539557488827564018, 4.63119879272535626664041260828, 5.41693810532010684114592369260, 6.58503742233429749451290233053, 7.30927763073737539225285904805, 7.978564281797096530197890335437
