| L(s) = 1 | − 4·7-s − 2·11-s + 13-s − 6·17-s − 4·19-s − 4·23-s + 6·29-s − 8·31-s + 10·37-s + 4·41-s − 4·43-s + 6·47-s + 9·49-s + 6·53-s − 6·59-s − 6·61-s + 10·71-s + 2·73-s + 8·77-s + 10·83-s − 8·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.624·41-s − 0.609·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.781·59-s − 0.768·61-s + 1.18·71-s + 0.234·73-s + 0.911·77-s + 1.09·83-s − 0.847·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.92239092938658, −14.34702855689569, −13.54361574944076, −13.40336050530056, −12.69504389226337, −12.58861366559551, −11.80316600115440, −11.11602422653268, −10.67531568661028, −10.22044784774745, −9.612812092727166, −9.094937836217174, −8.688312680555568, −7.978709690017578, −7.388078166196474, −6.729769966984974, −6.213457000972604, −5.984037499683875, −5.065523444591503, −4.342428571933606, −3.891001051585503, −3.155666737767904, −2.471774598783485, −2.017937026783280, −0.7121841432087288, 0,
0.7121841432087288, 2.017937026783280, 2.471774598783485, 3.155666737767904, 3.891001051585503, 4.342428571933606, 5.065523444591503, 5.984037499683875, 6.213457000972604, 6.729769966984974, 7.388078166196474, 7.978709690017578, 8.688312680555568, 9.094937836217174, 9.612812092727166, 10.22044784774745, 10.67531568661028, 11.11602422653268, 11.80316600115440, 12.58861366559551, 12.69504389226337, 13.40336050530056, 13.54361574944076, 14.34702855689569, 14.92239092938658
