| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s + 11-s − 4·13-s + 2·15-s + 4·17-s − 4·19-s − 8·21-s − 6·23-s + 25-s + 4·27-s + 10·29-s − 4·31-s − 2·33-s − 4·35-s − 2·37-s + 8·39-s − 10·41-s − 8·43-s − 45-s + 6·47-s + 9·49-s − 8·51-s − 2·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.970·17-s − 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.348·33-s − 0.676·35-s − 0.328·37-s + 1.28·39-s − 1.56·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s − 1.12·51-s − 0.274·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.249352180969310893124091064110, −7.43830406494792928247165218921, −6.68490218101470737292185331768, −5.86044811040013095840672728210, −4.97420459330455545206802104972, −4.74375816002078213350021356672, −3.68779393676913535536695451851, −2.33965017826349556217612695216, −1.28611294996796877603666741977, 0,
1.28611294996796877603666741977, 2.33965017826349556217612695216, 3.68779393676913535536695451851, 4.74375816002078213350021356672, 4.97420459330455545206802104972, 5.86044811040013095840672728210, 6.68490218101470737292185331768, 7.43830406494792928247165218921, 8.249352180969310893124091064110
