| L(s) = 1 | + 1.73·5-s + 7-s + 4.12·11-s − 7.04·13-s − 4.17·17-s + 19-s − 2.20·23-s − 1.98·25-s − 7.41·29-s + 3.42·31-s + 1.73·35-s + 10.1·37-s − 1.58·41-s + 3.37·43-s + 6.24·47-s + 49-s − 7.41·53-s + 7.15·55-s − 11.3·59-s − 2·61-s − 12.2·65-s − 0.236·67-s − 11.8·71-s + 5.47·73-s + 4.12·77-s + 2.20·79-s − 6.41·83-s + ⋯ |
| L(s) = 1 | + 0.776·5-s + 0.377·7-s + 1.24·11-s − 1.95·13-s − 1.01·17-s + 0.229·19-s − 0.459·23-s − 0.397·25-s − 1.37·29-s + 0.615·31-s + 0.293·35-s + 1.67·37-s − 0.248·41-s + 0.514·43-s + 0.910·47-s + 0.142·49-s − 1.01·53-s + 0.964·55-s − 1.47·59-s − 0.256·61-s − 1.51·65-s − 0.0289·67-s − 1.40·71-s + 0.640·73-s + 0.469·77-s + 0.248·79-s − 0.704·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 + 7.04T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 7.41T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 0.236T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 0.149T + 97T^{2} \) |
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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
−7.45625947878313979264116244704, −6.57612082139319197126820759531, −6.05611574777610614095585274357, −5.30105465212913633649013614478, −4.51041090818737744695743712946, −4.04774411960382690038134160734, −2.79530282636076119271684830503, −2.16580162953226382827997063400, −1.40533597019821998260745492059, 0,
1.40533597019821998260745492059, 2.16580162953226382827997063400, 2.79530282636076119271684830503, 4.04774411960382690038134160734, 4.51041090818737744695743712946, 5.30105465212913633649013614478, 6.05611574777610614095585274357, 6.57612082139319197126820759531, 7.45625947878313979264116244704
