L(s) = 1 | + (−2 + i)5-s + i·7-s − 4·11-s − 2i·13-s − 2i·17-s + 2·19-s − 6i·23-s + (3 − 4i)25-s + 6·29-s + 6·31-s + (−1 − 2i)35-s − 4i·37-s + 4i·43-s − 4i·47-s − 49-s + ⋯ |
L(s) = 1 | + (−0.894 + 0.447i)5-s + 0.377i·7-s − 1.20·11-s − 0.554i·13-s − 0.485i·17-s + 0.458·19-s − 1.25i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s + 1.07·31-s + (−0.169 − 0.338i)35-s − 0.657i·37-s + 0.609i·43-s − 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9671445661\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9671445661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.648307011387804568427389851268, −8.424041671504605213513899162462, −8.054098069079193760926880822262, −7.16116912953966265724109352289, −6.30159369237976561777077681965, −5.20569091422803011917605265448, −4.45778000480376715795849015311, −3.15298669682829436029692126461, −2.53401078227379528118069123077, −0.47043561184252440048275647383,
1.14909337484002361877807502805, 2.75155835323988551871707842161, 3.80552369448707893279649198732, 4.68324751356373303469515082095, 5.47333615347711159847509444142, 6.66379707164328041190353857707, 7.55514447012059832831887646127, 8.106167573825536348188629952611, 8.891353944728336320666834064006, 9.924142714344460158808893383846