| L(s) = 1 | + (−0.623 + 0.781i)2-s + (−1.14 + 2.92i)3-s + (−0.222 − 0.974i)4-s + (0.0747 + 0.997i)5-s + (−1.56 − 2.71i)6-s + (−0.830 + 1.43i)7-s + (0.900 + 0.433i)8-s + (−5.02 − 4.65i)9-s + (−0.826 − 0.563i)10-s + (0.278 − 1.22i)11-s + (3.10 + 0.467i)12-s + (−4.60 + 3.14i)13-s + (−0.606 − 1.54i)14-s + (−2.99 − 0.925i)15-s + (−0.900 + 0.433i)16-s + (−0.0106 + 0.142i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 + 0.552i)2-s + (−0.662 + 1.68i)3-s + (−0.111 − 0.487i)4-s + (0.0334 + 0.445i)5-s + (−0.640 − 1.10i)6-s + (−0.313 + 0.543i)7-s + (0.318 + 0.153i)8-s + (−1.67 − 1.55i)9-s + (−0.261 − 0.178i)10-s + (0.0839 − 0.367i)11-s + (0.895 + 0.135i)12-s + (−1.27 + 0.870i)13-s + (−0.162 − 0.413i)14-s + (−0.774 − 0.238i)15-s + (−0.225 + 0.108i)16-s + (−0.00258 + 0.0345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.237117 - 0.294561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.237117 - 0.294561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-3.63 + 5.45i)T \) |
| good | 3 | \( 1 + (1.14 - 2.92i)T + (-2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (0.830 - 1.43i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.278 + 1.22i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (4.60 - 3.14i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (0.0106 - 0.142i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-2.03 + 1.88i)T + (1.41 - 18.9i)T^{2} \) |
| 23 | \( 1 + (1.44 - 0.445i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (0.208 + 0.531i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (-7.70 - 1.16i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (2.84 + 4.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.97 - 7.48i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.754 - 3.30i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (8.00 + 5.45i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (8.79 - 4.23i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-2.30 + 0.346i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-9.64 + 8.94i)T + (5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (6.67 + 2.05i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (13.3 - 9.08i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (7.36 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.39 - 11.1i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (5.00 - 12.7i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 4.87i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.51292350719979254667701508818, −10.72254448907098368542120024487, −9.728119630084776412218073409445, −9.492819770127808861610492915220, −8.431106591113594406796068017140, −6.98620299157787927663102877406, −6.05198255782165383938513679527, −5.16021566500302311366945084784, −4.26419010026194453463742825938, −2.86347907066963970869209160295,
0.29908896945672072354125308571, 1.58434460811436847628126902764, 2.88360289478203061102466129475, 4.76096383621768460157666001262, 5.90421036768504985736594392875, 7.04466784254590571291282482752, 7.63156220551482383808192144706, 8.454603153819793822489967700405, 9.833084753332969074136644251104, 10.53046354920107059297656152818
