| L(s) = 1 | + 0.618·2-s + 3-s − 1.61·4-s + 3.23·5-s + 0.618·6-s − 2.23·8-s − 2·9-s + 2.00·10-s + 4.47·11-s − 1.61·12-s − 0.236·13-s + 3.23·15-s + 1.85·16-s − 1.23·18-s + 7.23·19-s − 5.23·20-s + 2.76·22-s − 23-s − 2.23·24-s + 5.47·25-s − 0.145·26-s − 5·27-s − 1.47·29-s + 2.00·30-s + 9·31-s + 5.61·32-s + 4.47·33-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 0.577·3-s − 0.809·4-s + 1.44·5-s + 0.252·6-s − 0.790·8-s − 0.666·9-s + 0.632·10-s + 1.34·11-s − 0.467·12-s − 0.0654·13-s + 0.835·15-s + 0.463·16-s − 0.291·18-s + 1.66·19-s − 1.17·20-s + 0.589·22-s − 0.208·23-s − 0.456·24-s + 1.09·25-s − 0.0286·26-s − 0.962·27-s − 0.273·29-s + 0.365·30-s + 1.61·31-s + 0.993·32-s + 0.778·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.629921525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.629921525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−9.600605435854286737054013622807, −9.104390229178503025244814392006, −8.529760048167120976518761716965, −7.30648981782828696847078997561, −6.08175246265480076408242380055, −5.69177561923566685682885455384, −4.65435797979836386906048392244, −3.54304263870856834504283365428, −2.66432946802710409905076536200, −1.27503629514242313524015434312,
1.27503629514242313524015434312, 2.66432946802710409905076536200, 3.54304263870856834504283365428, 4.65435797979836386906048392244, 5.69177561923566685682885455384, 6.08175246265480076408242380055, 7.30648981782828696847078997561, 8.529760048167120976518761716965, 9.104390229178503025244814392006, 9.600605435854286737054013622807
