| L(s) = 1 | − 3·3-s − 6.73·5-s + 9·9-s − 53.6·11-s + 1.73·13-s + 20.2·15-s − 46.9·17-s − 20.1·19-s + 118.·23-s − 79.6·25-s − 27·27-s − 103.·29-s − 157.·31-s + 161.·33-s − 37.7·37-s − 5.20·39-s − 287.·41-s + 504.·43-s − 60.6·45-s − 220.·47-s + 140.·51-s + 292.·53-s + 361.·55-s + 60.4·57-s − 595.·59-s − 265.·61-s − 11.6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.602·5-s + 0.333·9-s − 1.47·11-s + 0.0370·13-s + 0.347·15-s − 0.669·17-s − 0.243·19-s + 1.07·23-s − 0.636·25-s − 0.192·27-s − 0.663·29-s − 0.912·31-s + 0.849·33-s − 0.167·37-s − 0.0213·39-s − 1.09·41-s + 1.79·43-s − 0.200·45-s − 0.684·47-s + 0.386·51-s + 0.757·53-s + 0.886·55-s + 0.140·57-s − 1.31·59-s − 0.557·61-s − 0.0223·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6985655723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6985655723\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 6.73T + 125T^{2} \) |
| 11 | \( 1 + 53.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.73T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 595.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 936.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 545.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 940.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.418400607660161454095210893326, −8.512630975348776760785008683687, −7.63121245161861211513153774091, −7.05509481517871601602999263121, −5.91790213646041763819795529646, −5.15361212574179298074357278671, −4.30470578653869946203319952553, −3.19802616505864076812963507561, −2.00446306170654868220288999231, −0.42455003023466908121396932041,
0.42455003023466908121396932041, 2.00446306170654868220288999231, 3.19802616505864076812963507561, 4.30470578653869946203319952553, 5.15361212574179298074357278671, 5.91790213646041763819795529646, 7.05509481517871601602999263121, 7.63121245161861211513153774091, 8.512630975348776760785008683687, 9.418400607660161454095210893326
