| L(s) = 1 | − 1.41·7-s + 1.41·13-s + 6·17-s − 6·19-s − 8.48·23-s − 5·25-s + 8.48·29-s + 1.41·31-s + 7.07·37-s + 6·41-s − 6·43-s + 8.48·47-s − 5·49-s − 8.48·53-s − 7.07·61-s + 4·67-s + 8.48·71-s + 1.41·79-s − 12·83-s − 6·89-s − 2.00·91-s − 12·97-s + 8.48·101-s + 9.89·103-s − 12·107-s − 18.3·109-s + 6·113-s + ⋯ |
| L(s) = 1 | − 0.534·7-s + 0.392·13-s + 1.45·17-s − 1.37·19-s − 1.76·23-s − 25-s + 1.57·29-s + 0.254·31-s + 1.16·37-s + 0.937·41-s − 0.914·43-s + 1.23·47-s − 0.714·49-s − 1.16·53-s − 0.905·61-s + 0.488·67-s + 1.00·71-s + 0.159·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.21·97-s + 0.844·101-s + 0.975·103-s − 1.16·107-s − 1.76·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−7.51570025718327390381176918879, −6.36975596317611075848566902319, −6.23592645849099751279148135520, −5.45229323191015074898644921468, −4.40522038893178953404648831657, −3.91822001331664182388191006991, −3.04516474317418007342515769761, −2.23942291650490898635845795027, −1.21087440679347809127984064620, 0,
1.21087440679347809127984064620, 2.23942291650490898635845795027, 3.04516474317418007342515769761, 3.91822001331664182388191006991, 4.40522038893178953404648831657, 5.45229323191015074898644921468, 6.23592645849099751279148135520, 6.36975596317611075848566902319, 7.51570025718327390381176918879
