| L(s) = 1 | + 4i·2-s − 16·4-s − 32i·7-s − 64i·8-s − 12·11-s − 154i·13-s + 128·14-s + 256·16-s − 918i·17-s + 1.06e3·19-s − 48i·22-s + 4.22e3i·23-s + 616·26-s + 512i·28-s − 7.89e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.246i·7-s − 0.353i·8-s − 0.0299·11-s − 0.252i·13-s + 0.174·14-s + 0.250·16-s − 0.770i·17-s + 0.673·19-s − 0.0211i·22-s + 1.66i·23-s + 0.178·26-s + 0.123i·28-s − 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2655764326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2655764326\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 32iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 12T + 1.61e5T^{2} \) |
| 13 | \( 1 + 154iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 918iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.22e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.63e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 3.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.51e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.35e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.60e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.79e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.30e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.57e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.41e4iT - 8.58e9T^{2} \) |
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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
−10.78726855520332014540801123343, −9.576628099988045446863954518411, −9.137016443583774087552898915968, −7.64413347013860067126115027179, −7.44845839939376834691830498474, −6.06377910857388349559407130954, −5.30418448313549801016996393356, −4.15837645959130683235872925855, −3.00935145154580024072278212848, −1.31691786799994157736722790718,
0.06518207256015517363266000786, 1.46021803179062292323711403498, 2.59010208727547209907133030488, 3.73765720008763987763661476008, 4.78517523853562835039926027581, 5.87116454170814418918576109204, 6.97810192936610109666276224309, 8.223451748655338395694073579651, 8.900008099286484774872361142019, 9.969032478351885448809075079782
