| L(s) = 1 | + 2-s − 2.97·3-s + 4-s − 2.97·6-s + 3.57·7-s + 8-s + 5.85·9-s − 4.28·11-s − 2.97·12-s + 1.96·13-s + 3.57·14-s + 16-s + 6.37·17-s + 5.85·18-s − 7.08·19-s − 10.6·21-s − 4.28·22-s + 1.16·23-s − 2.97·24-s + 1.96·26-s − 8.48·27-s + 3.57·28-s + 29-s + 2.71·31-s + 32-s + 12.7·33-s + 6.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s − 1.21·6-s + 1.35·7-s + 0.353·8-s + 1.95·9-s − 1.29·11-s − 0.858·12-s + 0.546·13-s + 0.955·14-s + 0.250·16-s + 1.54·17-s + 1.37·18-s − 1.62·19-s − 2.32·21-s − 0.913·22-s + 0.243·23-s − 0.607·24-s + 0.386·26-s − 1.63·27-s + 0.675·28-s + 0.185·29-s + 0.487·31-s + 0.176·32-s + 2.21·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683185932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683185932\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 - 0.803T + 37T^{2} \) |
| 41 | \( 1 - 0.620T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 - 0.153T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−10.09914037516395238820248033050, −8.405035167716276615987420562365, −7.78528231232316504139911686492, −6.82385979515389081249231659682, −5.95440095317533158822670339633, −5.27342951208359110523971385183, −4.84464415925718854059552508541, −3.86269456731184878129446625254, −2.22655722682523895819667246304, −0.949892798736348240233440998415,
0.949892798736348240233440998415, 2.22655722682523895819667246304, 3.86269456731184878129446625254, 4.84464415925718854059552508541, 5.27342951208359110523971385183, 5.95440095317533158822670339633, 6.82385979515389081249231659682, 7.78528231232316504139911686492, 8.405035167716276615987420562365, 10.09914037516395238820248033050
