| L(s) = 1 | + 2-s + 2.19·3-s + 4-s + 3.16·5-s + 2.19·6-s − 7-s + 8-s + 1.83·9-s + 3.16·10-s + 5.88·11-s + 2.19·12-s − 1.85·13-s − 14-s + 6.95·15-s + 16-s + 2.64·17-s + 1.83·18-s + 0.189·19-s + 3.16·20-s − 2.19·21-s + 5.88·22-s − 7.48·23-s + 2.19·24-s + 5.01·25-s − 1.85·26-s − 2.55·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 1.41·5-s + 0.897·6-s − 0.377·7-s + 0.353·8-s + 0.612·9-s + 1.00·10-s + 1.77·11-s + 0.634·12-s − 0.513·13-s − 0.267·14-s + 1.79·15-s + 0.250·16-s + 0.642·17-s + 0.433·18-s + 0.0434·19-s + 0.707·20-s − 0.479·21-s + 1.25·22-s − 1.56·23-s + 0.448·24-s + 1.00·25-s − 0.363·26-s − 0.491·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.996639819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.996639819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 - 5.88T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 0.189T + 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 2.91T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 0.397T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.133268245134343348667957877376, −7.15401527390567298798874441777, −6.69883953605919498557850426637, −5.79771603180588099868296677690, −5.43168623723101907291271481140, −4.02605633349160424821599884953, −3.77715531212625684812574959321, −2.69621704812423707910896117148, −2.12884196910881552017747228160, −1.33843968688910358945479189299,
1.33843968688910358945479189299, 2.12884196910881552017747228160, 2.69621704812423707910896117148, 3.77715531212625684812574959321, 4.02605633349160424821599884953, 5.43168623723101907291271481140, 5.79771603180588099868296677690, 6.69883953605919498557850426637, 7.15401527390567298798874441777, 8.133268245134343348667957877376
