| L(s) = 1 | + (1.41 + 0.0443i)2-s + (−1.07 − 2.10i)3-s + (1.99 + 0.125i)4-s + (0.402 + 2.19i)5-s + (−1.42 − 3.02i)6-s + (0.707 + 0.707i)7-s + (2.81 + 0.265i)8-s + (−1.52 + 2.10i)9-s + (0.471 + 3.12i)10-s + (2.94 + 4.05i)11-s + (−1.87 − 4.34i)12-s + (0.838 + 5.29i)13-s + (0.968 + 1.03i)14-s + (4.20 − 3.21i)15-s + (3.96 + 0.500i)16-s + (−1.55 − 0.790i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0313i)2-s + (−0.620 − 1.21i)3-s + (0.998 + 0.0626i)4-s + (0.180 + 0.983i)5-s + (−0.581 − 1.23i)6-s + (0.267 + 0.267i)7-s + (0.995 + 0.0938i)8-s + (−0.508 + 0.700i)9-s + (0.149 + 0.988i)10-s + (0.888 + 1.22i)11-s + (−0.542 − 1.25i)12-s + (0.232 + 1.46i)13-s + (0.258 + 0.275i)14-s + (1.08 − 0.829i)15-s + (0.992 + 0.125i)16-s + (−0.376 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.60003 - 0.0599893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60003 - 0.0599893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0443i)T \) |
| 5 | \( 1 + (-0.402 - 2.19i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 3 | \( 1 + (1.07 + 2.10i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-2.94 - 4.05i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.838 - 5.29i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (1.55 + 0.790i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.27 + 3.90i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.15 + 7.31i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.15 - 1.02i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.33 + 2.05i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.31 - 1.15i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (7.79 + 5.66i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.87 + 1.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (9.67 - 4.92i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.97 + 3.55i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.83 - 1.33i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.28 + 6.02i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.23 + 6.35i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 0.422i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.0266 + 0.00421i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.40 - 10.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.74 + 2.41i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.16 - 1.60i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.89 - 11.5i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.87511639861754564124261762433, −9.806717521657668102987045623766, −8.482046102524994724085419203937, −7.10919793598788866012783128677, −6.68512669498251066628666285811, −6.46917407751000413899082375768, −5.00816304129025070821018501675, −4.07937556373659614386341654534, −2.45098698468310589877113231761, −1.73134064159466763734695777000,
1.26321366876838940030055706794, 3.37208727539680593001613868027, 4.02557353591377982128040913825, 5.09287095847222294218740151487, 5.56459384846962468120108833326, 6.40930340200419109218783292375, 7.962402096180443653176178147818, 8.720443105450755414665405359980, 10.04151170028376862518572507480, 10.42497967820143541482152668204
