| L(s) = 1 | + 2.18·2-s + 3-s + 2.79·4-s − 5-s + 2.18·6-s + 0.913·7-s + 1.73·8-s + 9-s − 2.18·10-s + 2.79·12-s + 4.37·13-s + 1.99·14-s − 15-s − 1.79·16-s − 1.73·17-s + 2.18·18-s + 3.46·19-s − 2.79·20-s + 0.913·21-s + 8.58·23-s + 1.73·24-s + 25-s + 9.58·26-s + 27-s + 2.55·28-s + 6.20·29-s − 2.18·30-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.39·4-s − 0.447·5-s + 0.893·6-s + 0.345·7-s + 0.612·8-s + 0.333·9-s − 0.692·10-s + 0.805·12-s + 1.21·13-s + 0.534·14-s − 0.258·15-s − 0.447·16-s − 0.420·17-s + 0.515·18-s + 0.794·19-s − 0.624·20-s + 0.199·21-s + 1.78·23-s + 0.353·24-s + 0.200·25-s + 1.87·26-s + 0.192·27-s + 0.481·28-s + 1.15·29-s − 0.399·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.096092782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.096092782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 7 | \( 1 - 0.913T + 7T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 6.20T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 7.02T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 6.10T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.961141728269512000933405639242, −8.567818598711140134378710843602, −7.43972974842292416070929871181, −6.77576189680028128036350222253, −5.90035377489134746741733962481, −4.93798597166886794200732303506, −4.34268128937694396451511026996, −3.36491837104066295491228602189, −2.85481575084439255847491670854, −1.39153051891853688674389288951,
1.39153051891853688674389288951, 2.85481575084439255847491670854, 3.36491837104066295491228602189, 4.34268128937694396451511026996, 4.93798597166886794200732303506, 5.90035377489134746741733962481, 6.77576189680028128036350222253, 7.43972974842292416070929871181, 8.567818598711140134378710843602, 8.961141728269512000933405639242
