L(s) = 1 | + 0.326·2-s − 1.89·4-s + 3.42·7-s − 1.27·8-s + 5.34·11-s − 3.52·13-s + 1.11·14-s + 3.37·16-s − 2.55·17-s − 2.02·19-s + 1.74·22-s − 7.57·23-s − 1.15·26-s − 6.48·28-s − 4.74·29-s + 1.62·31-s + 3.64·32-s − 0.835·34-s − 0.0134·37-s − 0.661·38-s − 9.67·41-s + 2.32·43-s − 10.1·44-s − 2.47·46-s − 6.94·47-s + 4.72·49-s + 6.66·52-s + ⋯ |
L(s) = 1 | + 0.231·2-s − 0.946·4-s + 1.29·7-s − 0.449·8-s + 1.61·11-s − 0.976·13-s + 0.299·14-s + 0.842·16-s − 0.620·17-s − 0.464·19-s + 0.372·22-s − 1.57·23-s − 0.225·26-s − 1.22·28-s − 0.880·29-s + 0.291·31-s + 0.644·32-s − 0.143·34-s − 0.00220·37-s − 0.107·38-s − 1.51·41-s + 0.354·43-s − 1.52·44-s − 0.364·46-s − 1.01·47-s + 0.674·49-s + 0.924·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 - 0.326T + 2T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 + 0.0134T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 - 0.0221T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 0.426T + 79T^{2} \) |
| 83 | \( 1 - 6.04T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.079076324302768645659927312737, −7.03924698409701099726082012019, −6.31262397793307371458183322321, −5.44876118426433355996237074982, −4.73621062207105718800269714658, −4.20918179062111219344780906730, −3.58006535812577732646540289733, −2.18332518224322208102246193960, −1.41486217621326607143574485847, 0,
1.41486217621326607143574485847, 2.18332518224322208102246193960, 3.58006535812577732646540289733, 4.20918179062111219344780906730, 4.73621062207105718800269714658, 5.44876118426433355996237074982, 6.31262397793307371458183322321, 7.03924698409701099726082012019, 8.079076324302768645659927312737