| L(s) = 1 | + 1.23·5-s + 7-s + 1.01·11-s + 3.64·13-s − 7.89·17-s − 19-s + 1.64·23-s − 3.47·25-s − 6.87·29-s − 2.65·31-s + 1.23·35-s + 1.37·37-s + 7.75·41-s − 3.28·43-s − 4.24·47-s + 49-s + 2.87·53-s + 1.25·55-s − 6.02·59-s − 5.28·61-s + 4.49·65-s − 1.01·67-s − 13.8·71-s + 13.7·73-s + 1.01·77-s − 6.08·79-s − 13.3·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 0.377·7-s + 0.305·11-s + 1.00·13-s − 1.91·17-s − 0.229·19-s + 0.342·23-s − 0.694·25-s − 1.27·29-s − 0.476·31-s + 0.208·35-s + 0.225·37-s + 1.21·41-s − 0.500·43-s − 0.619·47-s + 0.142·49-s + 0.395·53-s + 0.168·55-s − 0.784·59-s − 0.676·61-s + 0.558·65-s − 0.123·67-s − 1.64·71-s + 1.61·73-s + 0.115·77-s − 0.684·79-s − 1.46·83-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 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 61 | \( 1 + 5.28T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.25568216887724144284457922968, −6.63284110525862321314724853863, −5.96761976904012109483982311363, −5.44391833596170869814704851653, −4.39040318444003831963699206246, −4.03059357358809534955393642352, −2.95526377366830496479713995789, −2.02990286552915495894658217728, −1.43800571270927013686289373968, 0,
1.43800571270927013686289373968, 2.02990286552915495894658217728, 2.95526377366830496479713995789, 4.03059357358809534955393642352, 4.39040318444003831963699206246, 5.44391833596170869814704851653, 5.96761976904012109483982311363, 6.63284110525862321314724853863, 7.25568216887724144284457922968
