| L(s) = 1 | − 3·3-s − 30.4·7-s + 9·9-s − 31.4·11-s − 60.7·13-s − 121.·17-s + 14.4·19-s + 91.3·21-s − 13.6·23-s − 27·27-s − 76.0·29-s − 183.·31-s + 94.3·33-s − 37.3·37-s + 182.·39-s − 30.6·41-s − 327.·43-s + 449.·47-s + 583.·49-s + 363.·51-s − 301.·53-s − 43.3·57-s − 340.·59-s + 619.·61-s − 273.·63-s − 256.·67-s + 41.0·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.64·7-s + 0.333·9-s − 0.861·11-s − 1.29·13-s − 1.72·17-s + 0.174·19-s + 0.948·21-s − 0.124·23-s − 0.192·27-s − 0.486·29-s − 1.06·31-s + 0.497·33-s − 0.166·37-s + 0.748·39-s − 0.116·41-s − 1.16·43-s + 1.39·47-s + 1.70·49-s + 0.998·51-s − 0.782·53-s − 0.100·57-s − 0.752·59-s + 1.29·61-s − 0.547·63-s − 0.468·67-s + 0.0716·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1361202272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1361202272\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 30.4T + 343T^{2} \) |
| 11 | \( 1 + 31.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 256.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 499.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 19.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 914.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 521T + 9.12e5T^{2} \) |
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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
−9.523504360387039134612024273150, −8.739370632977086372810768079960, −7.41052576193236832772753119409, −6.91598415842556095753643954565, −6.05230932233422661370485244779, −5.20228698657375945226059444149, −4.23510767345722204067402401594, −3.07575325787667532296772635416, −2.14870256313771539122225034662, −0.17908911950567691482234778554,
0.17908911950567691482234778554, 2.14870256313771539122225034662, 3.07575325787667532296772635416, 4.23510767345722204067402401594, 5.20228698657375945226059444149, 6.05230932233422661370485244779, 6.91598415842556095753643954565, 7.41052576193236832772753119409, 8.739370632977086372810768079960, 9.523504360387039134612024273150
