| L(s) = 1 | + 25·5-s − 177.·7-s − 778.·11-s + 132.·13-s + 1.63e3·17-s + 345.·19-s + 959.·23-s + 625·25-s + 2.57e3·29-s + 4.12e3·31-s − 4.43e3·35-s + 7.18e3·37-s + 1.93e4·41-s + 1.52e3·43-s − 3.41e3·47-s + 1.46e4·49-s + 1.77e4·53-s − 1.94e4·55-s + 1.74e4·59-s − 1.84e4·61-s + 3.31e3·65-s − 5.41e4·67-s − 6.64e4·71-s − 2.89e4·73-s + 1.38e5·77-s − 8.80e4·79-s − 6.84e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.36·7-s − 1.93·11-s + 0.217·13-s + 1.37·17-s + 0.219·19-s + 0.378·23-s + 0.200·25-s + 0.568·29-s + 0.770·31-s − 0.611·35-s + 0.863·37-s + 1.79·41-s + 0.125·43-s − 0.225·47-s + 0.870·49-s + 0.868·53-s − 0.867·55-s + 0.652·59-s − 0.635·61-s + 0.0972·65-s − 1.47·67-s − 1.56·71-s − 0.635·73-s + 2.65·77-s − 1.58·79-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 + 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 778.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 132.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.63e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 345.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 959.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.93e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.78e4T + 8.58e9T^{2} \) |
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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
−8.786521000368248784809607982186, −7.82977658096385493135868915294, −7.13798897189126254363907005515, −5.96484926415612947480460256912, −5.58681221440291621648309813303, −4.39810570537217345453753940240, −2.98707134191599371575458763215, −2.73468382366069309734315184435, −1.07165719624001519837583142070, 0,
1.07165719624001519837583142070, 2.73468382366069309734315184435, 2.98707134191599371575458763215, 4.39810570537217345453753940240, 5.58681221440291621648309813303, 5.96484926415612947480460256912, 7.13798897189126254363907005515, 7.82977658096385493135868915294, 8.786521000368248784809607982186
