| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 2.85·7-s − 2.23·8-s − 11-s − 6.23·13-s + 4.61·14-s − 4.85·16-s − 0.618·17-s − 6.70·19-s − 1.61·22-s − 4.09·23-s − 10.0·26-s + 1.76·28-s + 1.38·29-s − 3·31-s − 3.38·32-s − 1.00·34-s − 10.2·37-s − 10.8·38-s + 3·41-s + 6·43-s − 0.618·44-s − 6.61·46-s + 11.9·47-s + 1.14·49-s − 3.85·52-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 1.07·7-s − 0.790·8-s − 0.301·11-s − 1.72·13-s + 1.23·14-s − 1.21·16-s − 0.149·17-s − 1.53·19-s − 0.344·22-s − 0.852·23-s − 1.97·26-s + 0.333·28-s + 0.256·29-s − 0.538·31-s − 0.597·32-s − 0.171·34-s − 1.68·37-s − 1.76·38-s + 0.468·41-s + 0.914·43-s − 0.0931·44-s − 0.975·46-s + 1.74·47-s + 0.163·49-s − 0.534·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 0.527T + 59T^{2} \) |
| 61 | \( 1 - 0.0901T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 + 1.61T + 97T^{2} \) |
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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
−8.566166564286067991311906282663, −7.64122594017502752818338039287, −6.95904042448582184642334706789, −5.89377592036249122182731729255, −5.26209711109087142135567756835, −4.51282209436054487755516122286, −4.03886761212716541143524476777, −2.68445124748225820137507666012, −2.01832159715590634393171950588, 0,
2.01832159715590634393171950588, 2.68445124748225820137507666012, 4.03886761212716541143524476777, 4.51282209436054487755516122286, 5.26209711109087142135567756835, 5.89377592036249122182731729255, 6.95904042448582184642334706789, 7.64122594017502752818338039287, 8.566166564286067991311906282663
