| L(s) = 1 | − 2.47·2-s − 0.974·3-s + 4.14·4-s + 2.41·6-s − 4.61·7-s − 5.30·8-s − 2.04·9-s − 4.98·11-s − 4.03·12-s − 1.91·13-s + 11.4·14-s + 4.86·16-s − 4.78·17-s + 5.07·18-s − 6.78·19-s + 4.50·21-s + 12.3·22-s + 3.91·23-s + 5.17·24-s + 4.75·26-s + 4.92·27-s − 19.1·28-s + 8.70·29-s − 7.90·31-s − 1.44·32-s + 4.85·33-s + 11.8·34-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.562·3-s + 2.07·4-s + 0.986·6-s − 1.74·7-s − 1.87·8-s − 0.683·9-s − 1.50·11-s − 1.16·12-s − 0.532·13-s + 3.05·14-s + 1.21·16-s − 1.16·17-s + 1.19·18-s − 1.55·19-s + 0.982·21-s + 2.63·22-s + 0.816·23-s + 1.05·24-s + 0.932·26-s + 0.947·27-s − 3.61·28-s + 1.61·29-s − 1.41·31-s − 0.255·32-s + 0.845·33-s + 2.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04932682404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04932682404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 + 0.974T + 3T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 1.28T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 + 8.31T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 3.78T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 + 0.588T + 73T^{2} \) |
| 79 | \( 1 + 7.26T + 79T^{2} \) |
| 83 | \( 1 - 2.15T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 5.14T + 97T^{2} \) |
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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
−9.919261377049362729134270999427, −8.992737281162218316031032954127, −8.540008553578724993007884230075, −7.45753298038215072847021121791, −6.61578844681179645048524978318, −6.16504674714590652465906606433, −4.88420636183669714911888691900, −3.04527528688178969100467095093, −2.33220451912539952333805150033, −0.20426832440579780939827814243,
0.20426832440579780939827814243, 2.33220451912539952333805150033, 3.04527528688178969100467095093, 4.88420636183669714911888691900, 6.16504674714590652465906606433, 6.61578844681179645048524978318, 7.45753298038215072847021121791, 8.540008553578724993007884230075, 8.992737281162218316031032954127, 9.919261377049362729134270999427
