| L(s) = 1 | − 2-s − 0.406·3-s + 4-s + 0.406·6-s − 2.91·7-s − 8-s − 2.83·9-s + 6.51·11-s − 0.406·12-s + 0.813·13-s + 2.91·14-s + 16-s + 2.51·17-s + 2.83·18-s + 0.406·19-s + 1.18·21-s − 6.51·22-s − 5.02·23-s + 0.406·24-s − 0.813·26-s + 2.37·27-s − 2.91·28-s − 5.32·29-s − 8.75·31-s − 32-s − 2.64·33-s − 2.51·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.234·3-s + 0.5·4-s + 0.166·6-s − 1.10·7-s − 0.353·8-s − 0.944·9-s + 1.96·11-s − 0.117·12-s + 0.225·13-s + 0.779·14-s + 0.250·16-s + 0.608·17-s + 0.668·18-s + 0.0933·19-s + 0.258·21-s − 1.38·22-s − 1.04·23-s + 0.0830·24-s − 0.159·26-s + 0.456·27-s − 0.551·28-s − 0.988·29-s − 1.57·31-s − 0.176·32-s − 0.460·33-s − 0.430·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9087307551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9087307551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 0.406T + 3T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 - 6.51T + 11T^{2} \) |
| 13 | \( 1 - 0.813T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 0.406T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 1.42T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
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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
−9.424295265597565237496197828071, −8.673781690543090470551272895382, −7.68809774822792529756549074244, −6.83240838993214284516969221567, −6.11817578520832340371691217898, −5.62688166349352375222505426732, −3.95374962816920303504898743368, −3.39682505186870354453442820794, −2.08484404725246553611869251207, −0.71886434868094370515394970061,
0.71886434868094370515394970061, 2.08484404725246553611869251207, 3.39682505186870354453442820794, 3.95374962816920303504898743368, 5.62688166349352375222505426732, 6.11817578520832340371691217898, 6.83240838993214284516969221567, 7.68809774822792529756549074244, 8.673781690543090470551272895382, 9.424295265597565237496197828071
