| L(s) = 1 | + 1.41·5-s + 0.414·7-s + 11-s − 2.41·13-s − 6.24·17-s − 5.24·19-s − 5.65·23-s − 2.99·25-s − 0.585·29-s + 3.65·31-s + 0.585·35-s − 3.82·37-s − 2.82·41-s − 11.6·43-s + 8.24·47-s − 6.82·49-s + 5.89·53-s + 1.41·55-s + 2.58·59-s − 7.58·61-s − 3.41·65-s + 8.31·67-s + 4·71-s + 10.8·73-s + 0.414·77-s + 0.0710·79-s + 9.65·83-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 0.156·7-s + 0.301·11-s − 0.669·13-s − 1.51·17-s − 1.20·19-s − 1.17·23-s − 0.599·25-s − 0.108·29-s + 0.656·31-s + 0.0990·35-s − 0.629·37-s − 0.441·41-s − 1.77·43-s + 1.20·47-s − 0.975·49-s + 0.810·53-s + 0.190·55-s + 0.336·59-s − 0.971·61-s − 0.423·65-s + 1.01·67-s + 0.474·71-s + 1.27·73-s + 0.0472·77-s + 0.00799·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 0.585T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 - 2.58T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 0.0710T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 0.242T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.574063726004378194210157928153, −7.974159435433534614337836303587, −6.78695147647228441367503700655, −6.42838059982853659724120688909, −5.43751900538719477984947112841, −4.57456324816818137344915036238, −3.80802053484926746956691956637, −2.40785679451189277094406812859, −1.83204467350320842831120968837, 0,
1.83204467350320842831120968837, 2.40785679451189277094406812859, 3.80802053484926746956691956637, 4.57456324816818137344915036238, 5.43751900538719477984947112841, 6.42838059982853659724120688909, 6.78695147647228441367503700655, 7.974159435433534614337836303587, 8.574063726004378194210157928153
