| L(s) = 1 | + 0.456·2-s − 3-s − 1.79·4-s − 0.456·6-s − 0.913·7-s − 1.73·8-s + 9-s + 1.79·12-s + 0.913·13-s − 0.417·14-s + 2.79·16-s − 7.02·17-s + 0.456·18-s − 5.29·19-s + 0.913·21-s − 0.582·23-s + 1.73·24-s + 0.417·26-s − 27-s + 1.63·28-s − 4.37·29-s + 2.58·31-s + 4.73·32-s − 3.20·34-s − 1.79·36-s + 9.58·37-s − 2.41·38-s + ⋯ |
| L(s) = 1 | + 0.323·2-s − 0.577·3-s − 0.895·4-s − 0.186·6-s − 0.345·7-s − 0.612·8-s + 0.333·9-s + 0.517·12-s + 0.253·13-s − 0.111·14-s + 0.697·16-s − 1.70·17-s + 0.107·18-s − 1.21·19-s + 0.199·21-s − 0.121·23-s + 0.353·24-s + 0.0818·26-s − 0.192·27-s + 0.309·28-s − 0.812·29-s + 0.463·31-s + 0.837·32-s − 0.550·34-s − 0.298·36-s + 1.57·37-s − 0.392·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5538653347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5538653347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 7 | \( 1 + 0.913T + 7T^{2} \) |
| 13 | \( 1 - 0.913T + 13T^{2} \) |
| 17 | \( 1 + 7.02T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 0.582T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 9.58T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 3.58T + 97T^{2} \) |
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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
−7.74243959180654127051225883832, −6.83613293358734418884000235450, −6.18551305191213546097860167630, −5.80574126195605281241223219465, −4.66354500626913801068804201298, −4.49305471040964725754929147441, −3.68074158028168216418707787581, −2.72736601934698820813231789579, −1.68296835685296249840624468093, −0.34892776674897581095998375712,
0.34892776674897581095998375712, 1.68296835685296249840624468093, 2.72736601934698820813231789579, 3.68074158028168216418707787581, 4.49305471040964725754929147441, 4.66354500626913801068804201298, 5.80574126195605281241223219465, 6.18551305191213546097860167630, 6.83613293358734418884000235450, 7.74243959180654127051225883832
