L(s) = 1 | + 1.23·3-s + 1.23·5-s − 1.47·9-s − 11-s + 3.23·13-s + 1.52·15-s − 2.47·17-s − 7.23·19-s − 4·23-s − 3.47·25-s − 5.52·27-s + 4.47·29-s + 2·31-s − 1.23·33-s + 6.94·37-s + 4.00·39-s + 2.47·41-s + 10.4·43-s − 1.81·45-s − 2·47-s − 3.05·51-s + 8.47·53-s − 1.23·55-s − 8.94·57-s + 2.76·59-s + 0.763·61-s + 4.00·65-s + ⋯ |
L(s) = 1 | + 0.713·3-s + 0.552·5-s − 0.490·9-s − 0.301·11-s + 0.897·13-s + 0.394·15-s − 0.599·17-s − 1.66·19-s − 0.834·23-s − 0.694·25-s − 1.06·27-s + 0.830·29-s + 0.359·31-s − 0.215·33-s + 1.14·37-s + 0.640·39-s + 0.386·41-s + 1.59·43-s − 0.271·45-s − 0.291·47-s − 0.427·51-s + 1.16·53-s − 0.166·55-s − 1.18·57-s + 0.359·59-s + 0.0978·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.60783299776523380222525090037, −6.62663483869870162383093394708, −5.99796601738700988588557087714, −5.62284073056579518662683769776, −4.26573243572770300763288217124, −4.07292430992864086471752383655, −2.76229801123229776836296643104, −2.43249307722473540471488866947, −1.45334134792195779769901679273, 0,
1.45334134792195779769901679273, 2.43249307722473540471488866947, 2.76229801123229776836296643104, 4.07292430992864086471752383655, 4.26573243572770300763288217124, 5.62284073056579518662683769776, 5.99796601738700988588557087714, 6.62663483869870162383093394708, 7.60783299776523380222525090037