L(s) = 1 | + 3.27·3-s + 1.56·5-s − 7-s + 7.72·9-s − 11-s − 13-s + 5.13·15-s − 3.96·17-s + 5.63·19-s − 3.27·21-s − 2.53·23-s − 2.54·25-s + 15.4·27-s + 8.75·29-s − 3.68·31-s − 3.27·33-s − 1.56·35-s + 6.52·37-s − 3.27·39-s − 2.06·41-s + 5.57·43-s + 12.1·45-s + 2.26·47-s + 49-s − 12.9·51-s + 10.8·53-s − 1.56·55-s + ⋯ |
L(s) = 1 | + 1.89·3-s + 0.700·5-s − 0.377·7-s + 2.57·9-s − 0.301·11-s − 0.277·13-s + 1.32·15-s − 0.962·17-s + 1.29·19-s − 0.714·21-s − 0.528·23-s − 0.509·25-s + 2.98·27-s + 1.62·29-s − 0.662·31-s − 0.570·33-s − 0.264·35-s + 1.07·37-s − 0.524·39-s − 0.323·41-s + 0.849·43-s + 1.80·45-s + 0.329·47-s + 0.142·49-s − 1.82·51-s + 1.49·53-s − 0.211·55-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
\(L(1)\) |
\(\approx\) |
\(5.023874183\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.023874183\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 6.52T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 0.931T + 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 - 9.98T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 6.84T + 73T^{2} \) |
| 79 | \( 1 + 1.30T + 79T^{2} \) |
| 83 | \( 1 + 5.68T + 83T^{2} \) |
| 89 | \( 1 + 7.44T + 89T^{2} \) |
| 97 | \( 1 - 8.98T + 97T^{2} \) |
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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
−7.927499855121745563703432917328, −7.28998433341531502680211226597, −6.68352086338453017628052747418, −5.80592177797580322195053650913, −4.85656725276156993953256597514, −4.07612379912543852744120683581, −3.37013776554983009935386978840, −2.50518601167129646199916838300, −2.20886128372682987377570286221, −1.04841440098051811227554984541,
1.04841440098051811227554984541, 2.20886128372682987377570286221, 2.50518601167129646199916838300, 3.37013776554983009935386978840, 4.07612379912543852744120683581, 4.85656725276156993953256597514, 5.80592177797580322195053650913, 6.68352086338453017628052747418, 7.28998433341531502680211226597, 7.927499855121745563703432917328