| L(s) = 1 | + (−0.540 + 0.841i)2-s + (2.56 − 0.368i)3-s + (−0.415 − 0.909i)4-s + (2.23 + 0.0141i)5-s + (−1.07 + 2.35i)6-s + (−1.82 − 1.58i)7-s + (0.989 + 0.142i)8-s + (3.56 − 1.04i)9-s + (−1.22 + 1.87i)10-s + (1.25 − 0.808i)11-s + (−1.40 − 2.17i)12-s + (−3.06 + 2.65i)13-s + (2.31 − 0.680i)14-s + (5.73 − 0.788i)15-s + (−0.654 + 0.755i)16-s + (−5.62 − 2.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (1.48 − 0.212i)3-s + (−0.207 − 0.454i)4-s + (0.999 + 0.00633i)5-s + (−0.439 + 0.962i)6-s + (−0.690 − 0.597i)7-s + (0.349 + 0.0503i)8-s + (1.18 − 0.348i)9-s + (−0.386 + 0.592i)10-s + (0.379 − 0.243i)11-s + (−0.404 − 0.629i)12-s + (−0.850 + 0.736i)13-s + (0.619 − 0.181i)14-s + (1.48 − 0.203i)15-s + (−0.163 + 0.188i)16-s + (−1.36 − 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61864 + 0.244691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61864 + 0.244691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-2.23 - 0.0141i)T \) |
| 23 | \( 1 + (-3.56 - 3.21i)T \) |
| good | 3 | \( 1 + (-2.56 + 0.368i)T + (2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (1.82 + 1.58i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 0.808i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.06 - 2.65i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.62 + 2.56i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.291 - 0.638i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.53 - 5.54i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.649 - 4.51i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 5.45i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (4.89 + 1.43i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.99 + 0.718i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 4.52iT - 47T^{2} \) |
| 53 | \( 1 + (4.01 + 3.48i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (9.64 + 11.1i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-2.04 + 14.2i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.77 - 2.75i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-9.51 - 6.11i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 1.39i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.54 - 1.77i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (0.823 + 2.80i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.515 + 3.58i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.34 - 11.3i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−12.72818333718737010491467697029, −11.00735655691083716467209755845, −9.677929303523433575743848166831, −9.351477915283201634831313071140, −8.492675294608260366602949972618, −7.10033498108854580242114102519, −6.68206612415911935994647047133, −4.94609555003827182336183009014, −3.31833838286434944346460610632, −1.91231656730616068047949782799,
2.21083232243369556706147168904, 2.86562216409464523109967665529, 4.36849800851549805395067320367, 6.08105274063563395054044976427, 7.45715266651279815883018166131, 8.692483330369834726762695137867, 9.280805583133769942420706682497, 9.861628390238204756396020308239, 10.90745818256319460014285891309, 12.47778769741653927707034255554
