| L(s) = 1 | + 34·5-s + 49·7-s − 340·11-s + 454·13-s + 798·17-s − 892·19-s − 3.19e3·23-s − 1.96e3·25-s + 8.24e3·29-s + 2.49e3·31-s + 1.66e3·35-s + 9.79e3·37-s − 1.98e4·41-s + 1.72e4·43-s + 8.92e3·47-s + 2.40e3·49-s − 150·53-s − 1.15e4·55-s − 4.23e4·59-s + 1.47e4·61-s + 1.54e4·65-s + 1.67e3·67-s + 1.45e4·71-s + 7.83e4·73-s − 1.66e4·77-s + 2.27e3·79-s − 3.77e4·83-s + ⋯ |
| L(s) = 1 | + 0.608·5-s + 0.377·7-s − 0.847·11-s + 0.745·13-s + 0.669·17-s − 0.566·19-s − 1.25·23-s − 0.630·25-s + 1.81·29-s + 0.466·31-s + 0.229·35-s + 1.17·37-s − 1.84·41-s + 1.42·43-s + 0.589·47-s + 1/7·49-s − 0.00733·53-s − 0.515·55-s − 1.58·59-s + 0.507·61-s + 0.453·65-s + 0.0456·67-s + 0.342·71-s + 1.72·73-s − 0.320·77-s + 0.0409·79-s − 0.601·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.623145715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623145715\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 34 T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 454 T + p^{5} T^{2} \) |
| 17 | \( 1 - 798 T + p^{5} T^{2} \) |
| 19 | \( 1 + 892 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3192 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8242 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2496 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19834 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17236 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8928 T + p^{5} T^{2} \) |
| 53 | \( 1 + 150 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42396 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14758 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1676 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 78378 T + p^{5} T^{2} \) |
| 79 | \( 1 - 2272 T + p^{5} T^{2} \) |
| 83 | \( 1 + 37764 T + p^{5} T^{2} \) |
| 89 | \( 1 - 117286 T + p^{5} T^{2} \) |
| 97 | \( 1 - 10002 T + p^{5} T^{2} \) |
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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
−9.273818411030066781341116711948, −8.224626150239023472888430959570, −7.83642042882463418611121368096, −6.49037953749424663674464240515, −5.88464503154097744342991914625, −4.95797894928742145215221689836, −3.97384780609277487090287047395, −2.76345639046587147916286646038, −1.84562466096342666353619371912, −0.70980824443123598372979772654,
0.70980824443123598372979772654, 1.84562466096342666353619371912, 2.76345639046587147916286646038, 3.97384780609277487090287047395, 4.95797894928742145215221689836, 5.88464503154097744342991914625, 6.49037953749424663674464240515, 7.83642042882463418611121368096, 8.224626150239023472888430959570, 9.273818411030066781341116711948
