| L(s) = 1 | − 25·5-s − 77.0·7-s − 403.·11-s + 858.·13-s − 1.39e3·17-s + 1.28e3·19-s − 794.·23-s + 625·25-s + 3.19e3·29-s + 6.56e3·31-s + 1.92e3·35-s + 2.92e3·37-s − 8.09e3·41-s − 4.96e3·43-s + 1.68e3·47-s − 1.08e4·49-s + 3.36e4·53-s + 1.00e4·55-s + 2.47e4·59-s − 4.06e4·61-s − 2.14e4·65-s + 4.84e4·67-s + 3.15e3·71-s + 1.22e4·73-s + 3.10e4·77-s + 4.82e4·79-s − 1.38e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.594·7-s − 1.00·11-s + 1.40·13-s − 1.17·17-s + 0.818·19-s − 0.313·23-s + 0.200·25-s + 0.704·29-s + 1.22·31-s + 0.265·35-s + 0.351·37-s − 0.751·41-s − 0.409·43-s + 0.111·47-s − 0.646·49-s + 1.64·53-s + 0.449·55-s + 0.925·59-s − 1.39·61-s − 0.630·65-s + 1.31·67-s + 0.0742·71-s + 0.269·73-s + 0.596·77-s + 0.868·79-s − 0.220·83-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 | \( 1 + 77.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 403.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 858.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 794.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.15e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.35e5T + 8.58e9T^{2} \) |
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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
−8.547261253624348532049595598371, −8.110428242544188080635449995156, −6.97059601334769093619726784392, −6.30188815971785205092080065741, −5.31237190615968215402257126357, −4.29821074484396991949575639402, −3.37095863083271136348863820647, −2.47793953749110019961414076510, −1.06192662101778802795079569996, 0,
1.06192662101778802795079569996, 2.47793953749110019961414076510, 3.37095863083271136348863820647, 4.29821074484396991949575639402, 5.31237190615968215402257126357, 6.30188815971785205092080065741, 6.97059601334769093619726784392, 8.110428242544188080635449995156, 8.547261253624348532049595598371
