| L(s) = 1 | + 2.30·3-s − 3.30·5-s − 7-s + 2.30·9-s − 2.60·11-s − 4·13-s − 7.60·15-s + 17-s − 2.60·19-s − 2.30·21-s − 6·23-s + 5.90·25-s − 1.60·27-s − 1.39·29-s + 5.90·31-s − 6·33-s + 3.30·35-s + 2.60·37-s − 9.21·39-s + 4.30·41-s − 6.30·43-s − 7.60·45-s − 5.21·47-s + 49-s + 2.30·51-s − 11.3·53-s + 8.60·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 1.47·5-s − 0.377·7-s + 0.767·9-s − 0.785·11-s − 1.10·13-s − 1.96·15-s + 0.242·17-s − 0.597·19-s − 0.502·21-s − 1.25·23-s + 1.18·25-s − 0.308·27-s − 0.258·29-s + 1.06·31-s − 1.04·33-s + 0.558·35-s + 0.428·37-s − 1.47·39-s + 0.671·41-s − 0.961·43-s − 1.13·45-s − 0.760·47-s + 0.142·49-s + 0.322·51-s − 1.55·53-s + 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 5.90T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 8.30T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 + 6.60T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 - 0.697T + 97T^{2} \) |
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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
−9.575000633797559668905173791678, −8.556816965355818168500355560242, −7.85714125387026698445005496734, −7.61362946964993857816982288803, −6.40433516545073710570374930734, −4.90988501592157583420722062542, −4.00071817373661143392219212010, −3.17142345248872428808923695964, −2.29326440631722086050783866146, 0,
2.29326440631722086050783866146, 3.17142345248872428808923695964, 4.00071817373661143392219212010, 4.90988501592157583420722062542, 6.40433516545073710570374930734, 7.61362946964993857816982288803, 7.85714125387026698445005496734, 8.556816965355818168500355560242, 9.575000633797559668905173791678
