L(s) = 1 | + 2.81·2-s − 6.54·3-s − 24.0·4-s + 45.9·5-s − 18.4·6-s − 157.·8-s − 200.·9-s + 129.·10-s − 551.·11-s + 157.·12-s − 1.09e3·13-s − 301.·15-s + 326.·16-s + 1.18e3·17-s − 563.·18-s + 1.16e3·19-s − 1.10e3·20-s − 1.55e3·22-s + 44.3·23-s + 1.03e3·24-s − 1.00e3·25-s − 3.07e3·26-s + 2.90e3·27-s + 3.32e3·29-s − 847.·30-s − 8.78e3·31-s + 5.96e3·32-s + ⋯ |
L(s) = 1 | + 0.497·2-s − 0.420·3-s − 0.752·4-s + 0.822·5-s − 0.209·6-s − 0.872·8-s − 0.823·9-s + 0.409·10-s − 1.37·11-s + 0.316·12-s − 1.79·13-s − 0.345·15-s + 0.318·16-s + 0.990·17-s − 0.409·18-s + 0.741·19-s − 0.618·20-s − 0.684·22-s + 0.0174·23-s + 0.366·24-s − 0.323·25-s − 0.893·26-s + 0.765·27-s + 0.735·29-s − 0.171·30-s − 1.64·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 - 2.81T + 32T^{2} \) |
| 3 | \( 1 + 6.54T + 243T^{2} \) |
| 5 | \( 1 - 45.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 551.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 44.3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 96.7T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.52T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e3T + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.05139146221272142200627163689, −12.92066105956800917297095500803, −11.99434569007563107093471435044, −10.31164073035846273470782939292, −9.364216183554720916865749405664, −7.73398342494379724287759921253, −5.67681985381235349729688342946, −5.07409854871435889407714482433, −2.81596682940432340996798615827, 0,
2.81596682940432340996798615827, 5.07409854871435889407714482433, 5.67681985381235349729688342946, 7.73398342494379724287759921253, 9.364216183554720916865749405664, 10.31164073035846273470782939292, 11.99434569007563107093471435044, 12.92066105956800917297095500803, 14.05139146221272142200627163689