L(s) = 1 | + (−1.24 − 0.674i)2-s + (−0.664 + 2.47i)3-s + (1.09 + 1.67i)4-s + (1.93 + 1.11i)5-s + (2.49 − 2.63i)6-s + (−0.224 − 2.81i)8-s + (−3.10 − 1.79i)9-s + (−1.65 − 2.69i)10-s + (−1.59 + 0.921i)11-s + (−4.88 + 1.58i)12-s + (2.94 − 2.94i)13-s + (−4.04 + 4.06i)15-s + (−1.62 + 3.65i)16-s + (−0.795 + 2.96i)17-s + (2.65 + 4.32i)18-s + (−2.66 + 4.61i)19-s + ⋯ |
L(s) = 1 | + (−0.878 − 0.476i)2-s + (−0.383 + 1.43i)3-s + (0.545 + 0.838i)4-s + (0.867 + 0.497i)5-s + (1.01 − 1.07i)6-s + (−0.0793 − 0.996i)8-s + (−1.03 − 0.597i)9-s + (−0.524 − 0.851i)10-s + (−0.481 + 0.277i)11-s + (−1.40 + 0.458i)12-s + (0.817 − 0.817i)13-s + (−1.04 + 1.05i)15-s + (−0.405 + 0.914i)16-s + (−0.192 + 0.720i)17-s + (0.625 + 1.01i)18-s + (−0.611 + 1.05i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.230607 + 0.794703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.230607 + 0.794703i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.24 + 0.674i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.664 - 2.47i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.59 - 0.921i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 + 2.94i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.795 - 2.96i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.66 - 4.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.44 + 0.654i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 6.35iT - 29T^{2} \) |
| 31 | \( 1 + (3.78 - 2.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.86 + 1.03i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + (1.02 + 1.02i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.210 + 0.785i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.61 + 0.699i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.11 - 3.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.00 + 10.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.19 + 0.856i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.5iT - 71T^{2} \) |
| 73 | \( 1 + (1.97 + 0.529i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.95 - 6.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.227 - 0.227i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.75 - 2.16i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.196 + 0.196i)T + 97iT^{2} \) |
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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
−10.38746657644624046235421427197, −9.817339556799845897393212114441, −8.901812939683384184461587538435, −8.235490751772837910391551652060, −6.96861838309991239488921633417, −6.00080662161796523593132917837, −5.15742103088773432243290264230, −3.83778091312747424167468797724, −3.11315598244682865797673128992, −1.71664405769892306392442687560,
0.53089807369242464448870473404, 1.65311292666505690732592885656, 2.51450129832012605216902325928, 4.79277199772077011485556741608, 5.77883584198855956053146429269, 6.43287906304286799642534992899, 7.02669558619866498761849860337, 7.962251748707826389565043494622, 8.770376129722406513777615271882, 9.379592463018758704295132279284