L(s) = 1 | − 2.41·3-s + 5-s + 2.82·9-s − 4.82·11-s + 0.828·13-s − 2.41·15-s − 0.828·17-s + 2.82·19-s + 2.41·23-s + 25-s + 0.414·27-s − 29-s + 6·31-s + 11.6·33-s − 1.99·39-s − 2.17·41-s − 6.41·43-s + 2.82·45-s − 2·47-s + 1.99·51-s − 6.82·53-s − 4.82·55-s − 6.82·57-s + 12.4·59-s − 11.4·61-s + 0.828·65-s − 12.4·67-s + ⋯ |
L(s) = 1 | − 1.39·3-s + 0.447·5-s + 0.942·9-s − 1.45·11-s + 0.229·13-s − 0.623·15-s − 0.200·17-s + 0.648·19-s + 0.503·23-s + 0.200·25-s + 0.0797·27-s − 0.185·29-s + 1.07·31-s + 2.02·33-s − 0.320·39-s − 0.339·41-s − 0.978·43-s + 0.421·45-s − 0.291·47-s + 0.280·51-s − 0.937·53-s − 0.651·55-s − 0.904·57-s + 1.62·59-s − 1.47·61-s + 0.102·65-s − 1.51·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.011720520931870932364051601660, −7.23746412904578506307064904724, −6.42166025279623473738336051477, −5.87669923672533610265851951844, −5.02901983418390223994746671800, −4.81675106584660287909935358319, −3.38553297403491952468088848846, −2.43507705103961987308389020965, −1.17177997092388157997457356948, 0,
1.17177997092388157997457356948, 2.43507705103961987308389020965, 3.38553297403491952468088848846, 4.81675106584660287909935358319, 5.02901983418390223994746671800, 5.87669923672533610265851951844, 6.42166025279623473738336051477, 7.23746412904578506307064904724, 8.011720520931870932364051601660