| L(s) = 1 | − 3-s − 3.61·5-s − 7-s + 9-s − 3.47·11-s − 4.61·13-s + 3.61·15-s − 17-s + 3·19-s + 21-s + 23-s + 8.09·25-s − 27-s + 1.47·29-s + 8.23·31-s + 3.47·33-s + 3.61·35-s − 7.47·37-s + 4.61·39-s + 8.70·41-s + 3.09·43-s − 3.61·45-s + 11.7·47-s + 49-s + 51-s − 5.61·53-s + 12.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.61·5-s − 0.377·7-s + 0.333·9-s − 1.04·11-s − 1.28·13-s + 0.934·15-s − 0.242·17-s + 0.688·19-s + 0.218·21-s + 0.208·23-s + 1.61·25-s − 0.192·27-s + 0.273·29-s + 1.47·31-s + 0.604·33-s + 0.611·35-s − 1.22·37-s + 0.739·39-s + 1.35·41-s + 0.471·43-s − 0.539·45-s + 1.70·47-s + 0.142·49-s + 0.140·51-s − 0.771·53-s + 1.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 3.09T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−7.45098938790068750769287639710, −7.09597146949090935399233946578, −6.14814326590784513029132957973, −5.25523732031202924586587783086, −4.68613136871519185753150400981, −4.06750945959632986704736704567, −3.10377390390626081455601276746, −2.47689636010037106252421986365, −0.829824697446790659867011047955, 0,
0.829824697446790659867011047955, 2.47689636010037106252421986365, 3.10377390390626081455601276746, 4.06750945959632986704736704567, 4.68613136871519185753150400981, 5.25523732031202924586587783086, 6.14814326590784513029132957973, 7.09597146949090935399233946578, 7.45098938790068750769287639710
