| L(s) = 1 | − 0.381·3-s − 3.85·7-s − 2.85·9-s − 11-s + 1.76·13-s + 1.61·17-s − 6.70·19-s + 1.47·21-s − 7.09·23-s + 2.23·27-s − 3.61·29-s + 3·31-s + 0.381·33-s + 5.76·37-s − 0.673·39-s − 3·41-s + 6·43-s + 5.94·47-s + 7.85·49-s − 0.618·51-s − 6.32·53-s + 2.56·57-s − 9.47·59-s − 11.0·61-s + 11.0·63-s − 8·67-s + 2.70·69-s + ⋯ |
| L(s) = 1 | − 0.220·3-s − 1.45·7-s − 0.951·9-s − 0.301·11-s + 0.489·13-s + 0.392·17-s − 1.53·19-s + 0.321·21-s − 1.47·23-s + 0.430·27-s − 0.671·29-s + 0.538·31-s + 0.0664·33-s + 0.947·37-s − 0.107·39-s − 0.468·41-s + 0.914·43-s + 0.867·47-s + 1.12·49-s − 0.0865·51-s − 0.868·53-s + 0.339·57-s − 1.23·59-s − 1.41·61-s + 1.38·63-s − 0.977·67-s + 0.326·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6978852988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6978852988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 0.854T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 0.618T + 97T^{2} \) |
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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
−8.302237848030734480471528634208, −7.76309679698595714328795093651, −6.64431581847945195977035837931, −6.10260919981987429322664830186, −5.76369652525753023369117691543, −4.54559173449320739024063198172, −3.71385126388493105999030899121, −2.96234805469970261260666499545, −2.09668634322180872240734963371, −0.44401108391812806689103824226,
0.44401108391812806689103824226, 2.09668634322180872240734963371, 2.96234805469970261260666499545, 3.71385126388493105999030899121, 4.54559173449320739024063198172, 5.76369652525753023369117691543, 6.10260919981987429322664830186, 6.64431581847945195977035837931, 7.76309679698595714328795093651, 8.302237848030734480471528634208
