| L(s) = 1 | + (−0.974 + 0.222i)2-s + (0.875 + 0.483i)3-s + (0.900 − 0.433i)4-s + (−0.600 − 1.08i)5-s + (−0.960 − 0.276i)6-s + (−0.275 − 0.345i)7-s + (−0.781 + 0.623i)8-s + (0.532 + 0.846i)9-s + (0.826 + 0.925i)10-s + (−5.63 + 1.97i)11-s + (0.998 + 0.0560i)12-s + (0.717 − 6.37i)13-s + (0.345 + 0.275i)14-s − 1.24i·15-s + (0.623 − 0.781i)16-s + (6.45 − 4.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.689 + 0.157i)2-s + (0.505 + 0.279i)3-s + (0.450 − 0.216i)4-s + (−0.268 − 0.485i)5-s + (−0.392 − 0.113i)6-s + (−0.104 − 0.130i)7-s + (−0.276 + 0.220i)8-s + (0.177 + 0.282i)9-s + (0.261 + 0.292i)10-s + (−1.69 + 0.594i)11-s + (0.288 + 0.0161i)12-s + (0.199 − 1.76i)13-s + (0.0924 + 0.0737i)14-s − 0.320i·15-s + (0.155 − 0.195i)16-s + (1.56 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.778311 - 0.554791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778311 - 0.554791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 3 | \( 1 + (-0.875 - 0.483i)T \) |
| 113 | \( 1 + (4.23 - 9.74i)T \) |
| good | 5 | \( 1 + (0.600 + 1.08i)T + (-2.66 + 4.23i)T^{2} \) |
| 7 | \( 1 + (0.275 + 0.345i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (5.63 - 1.97i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.717 + 6.37i)T + (-12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (-6.45 + 4.57i)T + (5.61 - 16.0i)T^{2} \) |
| 19 | \( 1 + (-2.27 + 0.654i)T + (16.0 - 10.1i)T^{2} \) |
| 23 | \( 1 + (0.670 - 2.32i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (4.27 + 6.01i)T + (-9.57 + 27.3i)T^{2} \) |
| 31 | \( 1 + (3.46 + 0.390i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (2.87 - 2.56i)T + (4.14 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-6.00 - 2.09i)T + (32.0 + 25.5i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 9.03i)T + (-40.5 - 14.2i)T^{2} \) |
| 47 | \( 1 + (5.66 - 0.318i)T + (46.7 - 5.26i)T^{2} \) |
| 53 | \( 1 + (2.03 + 4.21i)T + (-33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (0.483 + 1.67i)T + (-49.9 + 31.3i)T^{2} \) |
| 61 | \( 1 + (-2.83 - 8.09i)T + (-47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (-0.224 + 4.00i)T + (-66.5 - 7.50i)T^{2} \) |
| 71 | \( 1 + (-2.53 - 6.11i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-14.7 - 6.10i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.907 + 1.01i)T + (-8.84 - 78.5i)T^{2} \) |
| 83 | \( 1 + (2.03 + 8.92i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (4.34 - 0.738i)T + (84.0 - 29.3i)T^{2} \) |
| 97 | \( 1 + (-0.378 + 0.474i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.03629348691872141308166213695, −9.681641802255089750428782840853, −8.395779927917522077339329350975, −7.79460362214944115463010257616, −7.35504264528629047103245055690, −5.54513930052188492024030226013, −5.12424943430988930751209023309, −3.45650580599255508089290507092, −2.50935273023104330738354294599, −0.60312234223159366235845545065,
1.60129379661701171042866033913, 2.87870182280290631958879897794, 3.73496886236393171022555367324, 5.38206199001960059576848022169, 6.45059650184180757713748551176, 7.49350876251275912482062276505, 7.969012246801909842388912341425, 8.947633007362201833519390514539, 9.697596743752612757201160891230, 10.74986232199605597867061064319
