| L(s) = 1 | + 2·3-s − 4.37·5-s + 2.37·7-s + 9-s − 2·11-s + 4.74·13-s − 8.74·15-s − 2·17-s − 6.37·19-s + 4.74·21-s − 4.74·23-s + 14.1·25-s − 4·27-s − 8·29-s + 31-s − 4·33-s − 10.3·35-s − 4.74·37-s + 9.48·39-s − 4.37·41-s + 6.74·43-s − 4.37·45-s − 1.37·49-s − 4·51-s + 8.74·55-s − 12.7·57-s + 2.37·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.95·5-s + 0.896·7-s + 0.333·9-s − 0.603·11-s + 1.31·13-s − 2.25·15-s − 0.485·17-s − 1.46·19-s + 1.03·21-s − 0.989·23-s + 2.82·25-s − 0.769·27-s − 1.48·29-s + 0.179·31-s − 0.696·33-s − 1.75·35-s − 0.780·37-s + 1.51·39-s − 0.682·41-s + 1.02·43-s − 0.651·45-s − 0.196·49-s − 0.560·51-s + 1.17·55-s − 1.68·57-s + 0.308·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 4.37T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 0.744T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.391531556057010079022547473187, −8.231902224245010959284338617519, −7.60391080222722539824668755833, −6.69140968381002966650229120994, −5.41936179782201637667786844852, −4.10036081202971319653942681798, −4.01578564397495280769184225454, −2.93385672992749490704907857524, −1.79150980179495014933089088056, 0,
1.79150980179495014933089088056, 2.93385672992749490704907857524, 4.01578564397495280769184225454, 4.10036081202971319653942681798, 5.41936179782201637667786844852, 6.69140968381002966650229120994, 7.60391080222722539824668755833, 8.231902224245010959284338617519, 8.391531556057010079022547473187
