| L(s) = 1 | + 1.41·3-s + 4.24·7-s − 0.999·9-s − 5.65·11-s + 2·13-s − 6·17-s + 2.82·19-s + 6·21-s − 7.07·23-s − 5.65·27-s − 4·29-s + 2.82·31-s − 8.00·33-s − 2·37-s + 2.82·39-s + 8·41-s − 1.41·43-s − 1.41·47-s + 10.9·49-s − 8.48·51-s + 2·53-s + 4.00·57-s − 2.82·59-s − 14·61-s − 4.24·63-s − 4.24·67-s − 10.0·69-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 1.60·7-s − 0.333·9-s − 1.70·11-s + 0.554·13-s − 1.45·17-s + 0.648·19-s + 1.30·21-s − 1.47·23-s − 1.08·27-s − 0.742·29-s + 0.508·31-s − 1.39·33-s − 0.328·37-s + 0.452·39-s + 1.24·41-s − 0.215·43-s − 0.206·47-s + 1.57·49-s − 1.18·51-s + 0.274·53-s + 0.529·57-s − 0.368·59-s − 1.79·61-s − 0.534·63-s − 0.518·67-s − 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.66275934889948255428708706664, −7.47082863067393489542424340409, −6.07459062209976296324285287656, −5.53001457151267668643753643050, −4.70578023911069005922249711111, −4.11084221570304418649513339610, −2.99534230549887225721754859510, −2.30985958771953100426022479081, −1.63522986807116601983450718711, 0,
1.63522986807116601983450718711, 2.30985958771953100426022479081, 2.99534230549887225721754859510, 4.11084221570304418649513339610, 4.70578023911069005922249711111, 5.53001457151267668643753643050, 6.07459062209976296324285287656, 7.47082863067393489542424340409, 7.66275934889948255428708706664
