| L(s) = 1 | + 0.398·3-s + 3.53·5-s − 7-s − 2.84·9-s − 11-s − 13-s + 1.40·15-s − 1.07·17-s + 5.12·19-s − 0.398·21-s − 5.71·23-s + 7.46·25-s − 2.32·27-s − 1.17·29-s + 1.41·31-s − 0.398·33-s − 3.53·35-s − 7.80·37-s − 0.398·39-s − 5.27·41-s − 2.86·43-s − 10.0·45-s − 2.62·47-s + 49-s − 0.429·51-s − 6.09·53-s − 3.53·55-s + ⋯ |
| L(s) = 1 | + 0.229·3-s + 1.57·5-s − 0.377·7-s − 0.947·9-s − 0.301·11-s − 0.277·13-s + 0.362·15-s − 0.261·17-s + 1.17·19-s − 0.0868·21-s − 1.19·23-s + 1.49·25-s − 0.447·27-s − 0.218·29-s + 0.253·31-s − 0.0693·33-s − 0.596·35-s − 1.28·37-s − 0.0637·39-s − 0.824·41-s − 0.437·43-s − 1.49·45-s − 0.383·47-s + 0.142·49-s − 0.0601·51-s − 0.837·53-s − 0.476·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.398T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 + 1.98T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 8.91T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 2.00T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.47999279962735786814148748844, −6.62236974221753751616498953535, −6.08645914363649282098564586830, −5.40445970452034560379379648469, −5.00504435940209845845290836152, −3.70613522673688058138437502169, −2.94452849904266848381972655043, −2.27400241457980682878498697905, −1.49167266948210717582785077407, 0,
1.49167266948210717582785077407, 2.27400241457980682878498697905, 2.94452849904266848381972655043, 3.70613522673688058138437502169, 5.00504435940209845845290836152, 5.40445970452034560379379648469, 6.08645914363649282098564586830, 6.62236974221753751616498953535, 7.47999279962735786814148748844
