| L(s) = 1 | − 1.29·3-s + 1.75·5-s − 1.32·9-s + 0.221·11-s − 4.11·13-s − 2.27·15-s + 17-s + 0.347·19-s − 5.77·23-s − 1.91·25-s + 5.59·27-s + 8.59·29-s + 6.49·31-s − 0.286·33-s + 9.04·37-s + 5.32·39-s − 7.17·41-s + 6.05·43-s − 2.32·45-s + 7.97·47-s − 1.29·51-s − 11.5·53-s + 0.388·55-s − 0.449·57-s − 14.3·59-s − 6.78·61-s − 7.22·65-s + ⋯ |
| L(s) = 1 | − 0.747·3-s + 0.785·5-s − 0.441·9-s + 0.0666·11-s − 1.14·13-s − 0.586·15-s + 0.242·17-s + 0.0796·19-s − 1.20·23-s − 0.382·25-s + 1.07·27-s + 1.59·29-s + 1.16·31-s − 0.0498·33-s + 1.48·37-s + 0.852·39-s − 1.12·41-s + 0.922·43-s − 0.346·45-s + 1.16·47-s − 0.181·51-s − 1.58·53-s + 0.0523·55-s − 0.0595·57-s − 1.86·59-s − 0.868·61-s − 0.896·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 11 | \( 1 - 0.221T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 19 | \( 1 - 0.347T + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 - 6.05T + 43T^{2} \) |
| 47 | \( 1 - 7.97T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 4.88T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−7.76487155685859425038163572118, −6.61126736534399468798589339766, −6.24322863797328117641191797287, −5.59079353537328955099019018769, −4.87873523187374506710951469419, −4.24991816560940199915822744899, −2.93252627766610755819945670430, −2.37864894878597503587630066093, −1.20344731148356903019563190117, 0,
1.20344731148356903019563190117, 2.37864894878597503587630066093, 2.93252627766610755819945670430, 4.24991816560940199915822744899, 4.87873523187374506710951469419, 5.59079353537328955099019018769, 6.24322863797328117641191797287, 6.61126736534399468798589339766, 7.76487155685859425038163572118
