| L(s) = 1 | − 9·3-s + 24.6·5-s − 103.·7-s + 81·9-s − 457.·11-s − 1.02e3·13-s − 221.·15-s − 303.·17-s − 2.73e3·19-s + 928.·21-s − 529·23-s − 2.51e3·25-s − 729·27-s + 596.·29-s − 7.03e3·31-s + 4.12e3·33-s − 2.54e3·35-s − 5.33e3·37-s + 9.22e3·39-s − 1.26e3·41-s + 2.36e3·43-s + 1.99e3·45-s − 1.39e4·47-s − 6.16e3·49-s + 2.72e3·51-s + 1.22e4·53-s − 1.12e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.440·5-s − 0.795·7-s + 0.333·9-s − 1.14·11-s − 1.68·13-s − 0.254·15-s − 0.254·17-s − 1.73·19-s + 0.459·21-s − 0.208·23-s − 0.805·25-s − 0.192·27-s + 0.131·29-s − 1.31·31-s + 0.658·33-s − 0.350·35-s − 0.640·37-s + 0.971·39-s − 0.117·41-s + 0.195·43-s + 0.146·45-s − 0.923·47-s − 0.366·49-s + 0.146·51-s + 0.601·53-s − 0.502·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.003762203201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003762203201\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 24.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 457.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.02e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 303.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.73e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 596.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.11e4T + 8.58e9T^{2} \) |
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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
−9.327275733420502658491741576084, −8.268313851616527736571198661140, −7.29919228790836168061089921780, −6.61106607268203658534076394853, −5.68340052024844871135457519238, −5.00451477710872135859566172995, −3.98203224389988581836456012525, −2.64633320473433995395613996347, −1.94469843577102581133079840057, −0.02484588512520692912435549731,
0.02484588512520692912435549731, 1.94469843577102581133079840057, 2.64633320473433995395613996347, 3.98203224389988581836456012525, 5.00451477710872135859566172995, 5.68340052024844871135457519238, 6.61106607268203658534076394853, 7.29919228790836168061089921780, 8.268313851616527736571198661140, 9.327275733420502658491741576084
