L(s) = 1 | − 0.867i·3-s − 0.687i·5-s + (−2.28 − 1.33i)7-s + 2.24·9-s + (3.31 + 0.0124i)11-s − 13-s − 0.596·15-s + 3.72·17-s + 1.66·19-s + (−1.15 + 1.98i)21-s + 4.80·23-s + 4.52·25-s − 4.55i·27-s + 10.1i·29-s + 10.4i·31-s + ⋯ |
L(s) = 1 | − 0.500i·3-s − 0.307i·5-s + (−0.864 − 0.503i)7-s + 0.749·9-s + (0.999 + 0.00375i)11-s − 0.277·13-s − 0.154·15-s + 0.903·17-s + 0.381·19-s + (−0.251 + 0.432i)21-s + 1.00·23-s + 0.905·25-s − 0.875i·27-s + 1.88i·29-s + 1.87i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.108103507\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.108103507\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.28 + 1.33i)T \) |
| 11 | \( 1 + (-3.31 - 0.0124i)T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 0.867iT - 3T^{2} \) |
| 5 | \( 1 + 0.687iT - 5T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 - 10.1iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 - 0.528iT - 43T^{2} \) |
| 47 | \( 1 - 1.89iT - 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 + 3.88iT - 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 + 6.76iT - 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 - 7.61iT - 97T^{2} \) |
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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
−8.443070615703487959050999223705, −7.40128805083271359591358095943, −6.85508568964661768729034706838, −6.57343247088182049846456555073, −5.30356562327440412227470744290, −4.70890125816053690117660959591, −3.56410203837756363884992842075, −3.10977289501959587387615872761, −1.52506230081094326678910408230, −0.965112505258604148849815582984,
0.831598243006232181394486320906, 2.17689416414203689048431447148, 3.19717280267939892928021290635, 3.82999625651462603699630260831, 4.67040365523533530022764355584, 5.58056128541407690465541810050, 6.34180000055998663410440852270, 6.98253048623732052513731171509, 7.68282169313259598687721644938, 8.670537524881098339266606585341