L(s) = 1 | − 2.38·2-s + 3·3-s − 2.32·4-s + 14.5·5-s − 7.14·6-s + 22.4·7-s + 24.5·8-s + 9·9-s − 34.6·10-s + 11·11-s − 6.96·12-s + 58.9·13-s − 53.4·14-s + 43.5·15-s − 40.0·16-s + 12.2·17-s − 21.4·18-s − 99.5·19-s − 33.7·20-s + 67.3·21-s − 26.2·22-s + 41.8·23-s + 73.7·24-s + 86.0·25-s − 140.·26-s + 27·27-s − 52.0·28-s + ⋯ |
L(s) = 1 | − 0.842·2-s + 0.577·3-s − 0.290·4-s + 1.29·5-s − 0.486·6-s + 1.21·7-s + 1.08·8-s + 0.333·9-s − 1.09·10-s + 0.301·11-s − 0.167·12-s + 1.25·13-s − 1.02·14-s + 0.750·15-s − 0.625·16-s + 0.174·17-s − 0.280·18-s − 1.20·19-s − 0.376·20-s + 0.699·21-s − 0.254·22-s + 0.379·23-s + 0.627·24-s + 0.688·25-s − 1.05·26-s + 0.192·27-s − 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.938290409\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.938290409\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 2.38T + 8T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 13 | \( 1 - 58.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 47.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 1.36T + 2.97e4T^{2} \) |
| 37 | \( 1 - 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 417.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 52.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 621.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 749.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 351.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 820.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 449.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 985.T + 9.12e5T^{2} \) |
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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
−8.628071373901356043235113584479, −8.416973305320083809054064495267, −7.54981800613015870660268671248, −6.47972748864439319441105209403, −5.68099737957001659129419989366, −4.67273394314337846032073420031, −3.97108405592354180534018100330, −2.46921714181357104954406686259, −1.61834954732477811847931463857, −0.989802647859029715807972427924,
0.989802647859029715807972427924, 1.61834954732477811847931463857, 2.46921714181357104954406686259, 3.97108405592354180534018100330, 4.67273394314337846032073420031, 5.68099737957001659129419989366, 6.47972748864439319441105209403, 7.54981800613015870660268671248, 8.416973305320083809054064495267, 8.628071373901356043235113584479