| L(s) = 1 | − 4·2-s + 27.5·3-s + 16·4-s − 110.·6-s + 49·7-s − 64·8-s + 514.·9-s + 566.·11-s + 440.·12-s − 238.·13-s − 196·14-s + 256·16-s + 854.·17-s − 2.05e3·18-s + 1.84e3·19-s + 1.34e3·21-s − 2.26e3·22-s − 4.12e3·23-s − 1.76e3·24-s + 955.·26-s + 7.47e3·27-s + 784·28-s − 7.22e3·29-s + 4.90e3·31-s − 1.02e3·32-s + 1.55e4·33-s − 3.41e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s − 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.11·9-s + 1.41·11-s + 0.882·12-s − 0.391·13-s − 0.267·14-s + 0.250·16-s + 0.716·17-s − 1.49·18-s + 1.17·19-s + 0.667·21-s − 0.998·22-s − 1.62·23-s − 0.624·24-s + 0.277·26-s + 1.97·27-s + 0.188·28-s − 1.59·29-s + 0.916·31-s − 0.176·32-s + 2.49·33-s − 0.506·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.574094744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.574094744\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 27.5T + 243T^{2} \) |
| 11 | \( 1 - 566.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 238.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 854.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.84e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.60e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.38e5T + 8.58e9T^{2} \) |
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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
−10.20347647640354335138460066730, −9.410178445713308835027478743862, −8.990067507819972474415114932700, −7.73561182220425883398888906940, −7.55430450906983302458436309515, −6.06800771653844250880845935656, −4.24548235527092787267679474099, −3.31150454301165850009202475972, −2.11952965212645115672020278223, −1.16681719686071393847017741404,
1.16681719686071393847017741404, 2.11952965212645115672020278223, 3.31150454301165850009202475972, 4.24548235527092787267679474099, 6.06800771653844250880845935656, 7.55430450906983302458436309515, 7.73561182220425883398888906940, 8.990067507819972474415114932700, 9.410178445713308835027478743862, 10.20347647640354335138460066730
