L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s + i·8-s − 9-s + 3·11-s + 2i·12-s − i·13-s + 16-s + 6i·17-s + i·18-s − 19-s − 3i·22-s − 9i·23-s + 2·24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s + 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s − 0.277i·13-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s − 0.229·19-s − 0.639i·22-s − 1.87i·23-s + 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.221476552\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221476552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.550850246433368041702615164307, −7.79983141873359545484443776177, −6.92683881924668456693936582754, −6.30277971200210119303747218269, −5.45300789867343047372748065698, −4.20473625297410263714814984796, −3.58597229700582251951410040313, −2.19516494768558674776265842293, −1.66938604235428700514382012354, −0.41459514405894343044977775641,
1.50387593232407992736175011635, 3.20691277049366772666959683292, 3.88151128294011230237818422527, 4.71777690970606651764939440082, 5.34788663796958346446342040387, 6.21532830151476193457497773795, 7.19545151913731595438329629101, 7.66663578946429618992588832324, 8.921303883531473803900210860192, 9.434853445333141616286163736917